text:
A Research Agenda for Assessment and Management of Psychosis in Emergency Department Patients
Articles , Behavioral Health , Current Issue: Volume 20 Issue 2 , Review Article
DOI: 10.5811/westjem.2019.1.39263
Jennifer Peltzer-Jones, PsyD, RN Henry Ford Health System, Department of Emergency Medicine, Detroit, Michigan
Kimberly Nordstrom, MD, JD University of Colorado School of Medicine, Department of Psychiatry, Denver, Colorado
Glenn Currier, MD, MPH University of South Florida, Morsani College of Medicine, Department of Psychiatry, Tampa, Florida
Jon S. Berlin, MD Medical College of Wisconsin, Department of Psychiatry and Behavioral Medicine, Department of Emergency Medicine, Milwaukee, Wisconsin
Cynthia Singh, MS American College of Emergency Physicians, Irving, Texas
Sandra Schneider, MD American College of Emergency Physicians, Irving, Texas;
John Peter Smith Hospital, Department of Emergency Medicine, Fort Worth, Texas;
Hofstra Northwell School of Medicine, Hempstead, New York
Emergency departments (ED) manage a wide variety of critical medical presentations. Traumatic, neurologic, and cardiac crises are among the most prevalent types of emergencies treated in an ED setting. The high volume of presentations has led to collaborative partnerships in research and process development between experts in emergency medicine (EM) and other disciplines. While psychosis is a medical emergency frequently treated in the ED, there remains a paucity of evidence-based literature highlighting best practices for management of psychotic presentations in the ED. In the absence of collaborative research, development of best practice guidelines cannot begin. A working group convened to develop a set of high-priority research questions to address the knowledge gaps in the care of psychotic patients in the ED. This article is the product of a subgroup considering "Special Populations: Psychotic Spectrum Disorders," from the 2016 Coalition on Psychiatric Emergencies first Research Consensus Conference on Acute Mental Illness.
Participants were identified with expertise in psychosis from EM, emergency psychiatry, emergency psychology, clinical research, governmental agencies, and patient advocacy groups. Background literature reviews were performed prior to the in-person meeting. A nominal group technique was employed to develop group consensus on the highest priority research gaps. Following the nominal group technique, input was solicited from all participants during the meeting, questions were iteratively focused and revised, voted on, and then ranked by importance.
The group developed 28 separate questions. After clarification and voting, the group identified six high-priority research areas. These questions signify the perceived gaps in psychosis research in emergency settings. Questions were further grouped into two topic areas: screening and identification; and intervention and management strategies.
While psychosis has become a more common presentation in the ED, standardized screening, intervention, and outcome measurement for psychosis has not moved beyond attention to agitation management. As improved outpatient-intervention protocols are developed for treatment of psychosis, it is imperative that parallel protocols are developed for delivery in the ED setting.
Psychosis is an important clinical problem, not only for patients but for families and healthcare workers as well. Patients with mental disorders represent an increasing fraction of total presentations to emergency departments (ED) over time. In 2014, mental disorders were the 10th leading cause of United States ED visits for males aged 15–65 years, and mental disorders were the primary ED diagnosis in slightly over five million ED visits.1 Thus, development of better management approaches to assess and treat psychosis has become critical. With other high volume/high risk medical emergencies – traumatic injuries, cerebrovascular accidents, cardiac arrhythmias – emergency medicine (EM) has been able to partner with other medical specialties to jointly research and develop best practice care. However, translation of best practice care of psychosis specific to an emergency setting has yet to occur.
Mounting evidence suggests early intervention predicts improved outcomes in younger, first-episode psychotic patients. Yet to our knowledge, no evidence-based interventions linking first-episode psychotic ED patients into specialized treatment have been tested. This deficit highlights the collaborative treatment chasm between mental health and ED specialty fields. Appropriate recognition, categorization and management of psychosis should be a key element of comprehensive emergency care; achieving these goals can be done through improved mental health and emergency care collaboration. The goal of this research consensus workgroup was to explore and enumerate the current knowledge gaps for the care of psychosis specifically in an emergency setting.
Participants from a variety of disciplines – EM, emergency psychiatry, emergency psychology, clinical research, governmental agencies, and patient advocacy groups – were invited to participate in a research consensus session held prior to a joint emergency-psychiatry conference (the 7th Annual National Update on Behavioral Emergencies). Background literature reviews were performed prior to the in-person meeting. Literature reviews were conducted via journal review, academic databases and web-based searches. Searches fell within the scope of the priority domain identified by the Coalition on Psychiatric Emergencies (CPE) steering committee: acute psychosis. The workgroup leaders identified articles of importance and circulated them electronically to the group for review in advance of the in-person meeting. A nominal group technique2 was employed to develop group consensus on the highest priority research gaps. Following the nominal group technique, input was solicited from all participants during the meeting, questions were iteratively focused and revised, voted on, and then ranked by importance. Following the in-person meeting, the workgroup developed additional consensus and worked electronically to further refine the final form of each question. Please see the Executive Summary for the full methods (Appendix).
The group consisted of three emergency psychiatrists, an emergency psychologist, an emergency physician, clinical researcher and participant from a professional medical association. The average age of the participants was approximately 40 years old and included five females and two males. The group developed 28 separate questions. After clarification and voting, the group identified six high-priority research areas. Questions were further grouped into two topic areas: screening and identification; and intervention and management strategies. The questions organized by topic, are included in Table 1 and 2.
Table 1Key research questions to guide efforts for individuals with psychosis through screening and identification.
Question 1 Can a research-based triage tool be developed to assess psychosis in ED patients?
Question 2 What outcomes are meaningful for patients/families when assessing the effectiveness of psychosis interventions?
ED, emergency department.
Table 2Key research questions to guide efforts for effective intervention and management of the patient with acute psychosis.
Question 3 What is the recommended treatment for psychosis in the emergency setting?
Question 4 What affects emergency provider decision-making in treatment choice for psychosis?
Question 5 What system outcomes can be affected by early treatment of psychosis in emergency settings – both within the emergency care setting and thereafter?
Question 6 Are there appropriate care locations for psychotic patient presentations instead of the ED?
This discussion highlights current knowledge gaps and rationale as to why improved patient care processes cannot be implemented until this research is conducted.
Question 1: Can a research-based triage tool be developed to assess psychosis in ED patients?
Psychosis is a symptom rather than a definitive diagnosis, and it is a continuous rather than a categorical phenomenon. At one extreme, patients can be quietly delusional and at risk of self-harm and at the other extreme, paranoid with poor reality testing posing an extreme and immediate risk to ED staff and other patients. While rating scales for psychosis such as the Brief Psychiatric Rating Scale have been employed for over 50 years and assess several domains of psychosis, they have not been incorporated into ED care.3 The ED-based scales that have been developed and normed tend to focus primarily on agitation as the primary outcome and not on the variety of psychosis symptoms.
Tools assessing positive and negative domains of psychosis have not been standardized as valid or reliable within an ED setting.4 This deficit has led to a misunderstanding of the true incidence and prevalence of psychosis presentations in EDs. Without better, clearer definitions, ED and mental health providers will continue to have a chasm in care. Articulation of a more refined definition of psychosis that is measurable, relevant to ED care, understandable to ED providers, and captures the most salient symptoms of psychosis should be a high priority on any research agenda to ensure that both mental health and emergency providers are sharing a common language. Creation of such tools can then guide goal-directed treatment strategies within the ED.
An additional difficulty with psychosis presentations to an ED is the heterogeneous etiologies that can produce episodes of psychoses. Psychotic presentations are not all related to underlying mental illness (e.g., postictal states, metabolic derangements, substance intoxication/withdrawal, etc.). Because of this, there is a need to employ organized evaluations of psychosis. Using standardized algorithms would improve correct etiology identification and lead to proper treatment choices. For example, up to half of patients presenting to psychiatric EDs have concurrent substance use disorder. Schanzer et al. found ED clinicians inaccurately ascribed first presentations of psychosis to primary psychiatric disorders instead of substance misuse in one quarter of patients evaluated.5 This type of inaccurate diagnosing creates missed opportunities for chemical dependency interventions and leads to referral of patients to the wrong levels of care. Standardized ED medical evaluation algorithms for psychosis have been published in academic literature and adopted in several states, which help ED staff detect primary psychotic disorders from medical mimics.6-11 Without universal adaptation of medical evaluation protocols for psychotic presentations, there is a continued risk of misidentifying mental health etiology from medical etiology, leading to inappropriate or missed interventions.
Question 2: What outcomes are meaningful for patients/families when assessing the effectiveness of psychosis interventions?
At present, literature regarding emergent psychosis intervention has predominantly focused on management of agitation.12-15 First- or second-generation antipsychotic medication interventions have measured outcomes such as achieving calm behavior16 or decreasing need for additional medications.15,17 These measures neglect the vast spectrum of distressing, patient-level experiences of psychosis such as delusional thought content, sensory hallucinations, and negative affective states. While agitation can be a symptom of psychosis, agitation is not a pathognomonic symptom for psychosis; thus, efficacy of psychosis interventions must be broadened. While it is possible there is a direct link between treatment of agitation and alleviation of patient symptoms, further research in this field is needed. Because the bulk of literature has focused on management of agitation, it is not well known what the most important outcomes are for psychosis intervention in the ED according to patients and families. The effects of emergency intervention care choices relative to patient/family satisfaction, patient quality of life, patient course of illness, future patient/family crisis help seeking, etc., is also largely unknown. Additional patient- and family-centered studies in this area are necessary.
Question 3: What is the recommended treatment for psychosis in the emergency setting?
There is mounting evidence that early and aggressive intervention for first-episode psychosis (FEP) related to schizophrenia makes a significant impact on longer term outcomes.18 Since many patients with FEP present initially to the ED rather than to mental health treatment settings, opportunities to link patients into care are dependent upon the knowledge base of the ED providers. As compared to other medical disorders treated in the emergency setting, there is a significant deficit in best practice interventions for first, or subsequent, episodes of psychosis. At least one randomized, controlled trial demonstrated the superiority of outpatient, multimodal treatment strategies for FEP as compared to treatment as usual,19 but how similar interventions can be developed for an emergency setting is unclear. More specifically, while recommendations for psychosis treatment are available in psychiatric literature, no studies have yet standardized the education and engagement of these non-ED best practice recommendations such as medication management, family psychoeducation, social skills training, and supported employment/education programs, into emergency care protocols.19,20 Therefore, it is not known if rapid linkage to specialized outpatient treatment can improve outcomes.
It could be argued that the lack of standardized algorithms for new onset psychosis care as compared to interventions for other newly diagnosed disease states, such as diabetes, represents both a healthcare disparity in how mental illness is managed and a chasm in collaborative care between emergency and mental health researchers. As programs for earlier identification and intervention (i.e., prodromal presentations) are implemented nationally and internationally, it is not well defined as to how emergency providers will receive education and training to identify individuals at risk and provide recommended care.21 In addition to management of FEP, it is unclear what best practice emergency guidelines are for psychosis decompensation along the life course of the illness. It is not known if psychotic presentations in the first three years of an illness should be targeted and treated differently than in later years.
Aside from medication strategies, little research has been conducted investigating non-pharmacologic interventions for psychosis in the ED. Psychotic patients in the ED have a wide variety of behavioral presentations, often with subtle but important variations. For example, agitated patients may self-present seeking appropriate and effective medication for their condition, or they may be brought in involuntarily because of resistance to treatment, hostility, paranoia, and physical aggression. Often the literature on psychotic agitation does not distinguish between these two presentations and focuses on selecting an appropriate medication and route of medication for agitation. However, the importance of engagement, collaboration and, specifically, the art of engaging the individual around medication is key.22
Psychiatric emergency service (PES) practitioners note a significant reduction in outcomes such as decreased use of restraint and seclusion, as well as increased safety to both staff and patient, when the attempt to form a therapeutic alliance is prioritized.23 PES refers to specialized psychiatric crisis response centers and are not housed within EDs, generally managed by psychiatrically trained staff. ED providers may not receive the same training on building therapeutic alliances with patients as compared to mental health practitioners. It is unclear if providing increased education to ED providers on enhancing patient alliance could lead to improved ED patient engagement, as these types of outcome studies have not been conducted.
Question 4: What affects emergency provider decision-making in treatment choice for psychosis?
In a recent longitudinal review, Bessaha's group highlighted the lack of standardized clinical protocols when they examined disposition decisions for psychotic illness presentations.24 There were significant differences in hospitalization rates dependent upon non-clinical factors such as race, gender, and geographic location, although why these differences exist is unknown. How patients present to emergency settings, what resources are available to them, the level of emergency provider training in behavioral health assessment, and familiarity with psychopharmacology principles all may ultimately contribute to the disposition decision-making of the emergency provider. It is not understood how the interplay between patient severity level and non-patient factors combines to determine treatment decisions. It is unclear if these decisions are efficacious in illness management.
Question 5: What system outcomes can be affected by early treatment of psychosis in emergency settings – both within emergency care settings and thereafter?
While earlier questions focused on patient-centered outcomes, it is not known if evidenced-based care can positively affect system-level outcomes such as ED throughput. Nationally, there is recognition that patients with mental health complaints have longer ED lengths of stay (LOS) than those presenting without mental health complaints.25 More specifically, patients who present in mental health crisis and who have a diagnosis of psychosis have longer ED LOS than patients without mental health complaints.16,26 At present, knowledge gaps exist in how often a patient receives an intradepartmental intervention, how early into an emergency presentation patients receive treatment, and whether earlier intradepartmental interventions can make a difference in disposition choices. These metrics are not monitored in the same way EDs deliver interventions such as early goal-directed treatment of sepsis, time to cardiac catheterization, or door to needle time for cerebrovascular accidents. Creating evidenced-based guidelines and metrics for acute mental illness should mimic acute medical disorder protocols. Because we do not have a standard, goal-directed psychosis treatment algorithm, it is unclear if early treatment can affect ED throughput, subsequent inpatient psychiatric LOS, or safety outcomes (i.e., use of restraints/seclusion or patient/family/provider injury).
Question 6: Are there appropriate care locations for psychotic patient presentations instead of the ED?
With increasing alternative models of care – specifically PES – it is not fully known how these settings can contribute to better patient or system outcomes. Mental health systems of care do not have standardized formulas on which to base decisions about developing new facilities, and PES are not all developed and accessed in the same way. How PES care enhances psychosis management differently than general ED care as it relates to patient- and system-level outcomes is unknown. For example, in comparing PES services with general EDs, which site provides more consistent psychosis interventions, which site is better able to serve first-onset psychosis vs safety net concerns such as medication refills; which site works better with non-mental health professionals (such as emergency medical services, or police)? Additional research is needed to compare and contrast psychosis outcomes between these differing models of care.
There are several limitations to this study. First, this was not an empirical literature review, but rather an expert group of research clinicians and others who engaged in a nominal group technique to come to a consensus on setting future research priorities for the management of psychosis in the ED based on the knowledge of the current gaps in existing literature. Due to the lack of existing literature on psychosis management in the ED, the two articles sent for review prior to the conference focused on early interventions for psychosis in the community setting.19, 21 By the time this paper is published, it is possible studies may have been conducted that focus on the gaps in knowledge outlined through this research consensus conference. An additional limitation includes use of the nominal group technique, as it is different from large literature reviews/meta-analytic studies, which highlight what is known. This meeting and subsequent discussion focused on gaps in literature in order to set a future research agenda focused on psychosis management in the ED. Discussing what does not exist vs what is known could be perceived as a limitation. Our hope is that in highlighting what is missing from current literature, we can help shape research agendas moving forward.
Another limitation was in the psychosis workgroup selection. While the group engaged a variety of practitioners from emergency settings, it was limited to emergency specialists. One could argue that the inclusion of important stakeholders, such as inpatient psychiatric clinicians, could have provided additional perspectives on what areas are of highest priority to explore. Lastly, the group focus was narrowed to primary psychosis and did not include psychotic presentations due to substance intoxication/withdrawal or medical etiologies. We excluded substance-related psychosis presentations because we knew that a different group at this conference, which focused on substance-related presentations, was performing an identical critical review. Identification of psychotic presentations due to underlying medical problems has been extensively discussed in the literature in the context of the ongoing medical clearance work. The group felt it was of greater impact to focus on primary psychotic illness management, which has not had the same type of attention and focus in the research literature.
EDs are increasingly expected to provide interventions for acute psychosis, both for first episodes of psychosis or during exacerbations of chronic illness, yet there are no current, evidence-based protocols for treatment of psychosis care. Addressing the identified research questions would serve as first steps in developing standardized algorithms for psychosis care and improving treatment in the ED setting.
Section Editor: Marc L. Martel, MD
Address for Correspondence: Jennifer Peltzer-Jones, PsyD, Henry Ford Health System, Department of Emergency Medicine, 2799 W. Grand Blvd, Detroit, MI 48202. Email: jpeltze1@hfhs.org. 3 / 2019; 20:403 – 408
Submission history: Revision received June 4, 2018; Submitted January 29, 2019; Accepted January 29, 2019
Conflicts of Interest: By the WestJEM article submission agreement, all authors are required to disclose all affiliations, funding sources and financial or management relationships that could be perceived as potential sources of bias. The Coalition is funded, in part, by an unrestricted grant from Alexza.
1. National Center for Health Statistics. Available at: https://www.cdc.gov/nchs/data/nhamcs/web_tables/2014_ed_web_tables.pdf. Accessed November 11, 2017.
2. Gaining consensus among stakeholders through the nominal group technique. 2006. Available at: https://www.cdc.gov/healthyyouth/evaluation/pdf/brief7.pdf. Accessed December 1, 2016.
3. Overall JE, Gorham DR. The brief psychiatric rating scale. Psychol Rep. 1962;10(3):799-812.
4. Montoya A, Valladares A, Lizan L, et al. Validation of the Excited Component of the Positive and Negative Syndrome Scale (PANSS-EC) in a naturalistic sample of 278 patients with acute psychosis and agitation in a psychiatric emergency room. Health Qual Life Outcomes. 2011;9:18.
5. Schanzer BM, First MB, Domingues B, et al. Diagnosing psychotic disorders in the emergency department in the context of substance use. Psychiatr Serv. 2006;57(10):1468-73.
6. Lukens TW, Wolf SJ, Edlow JA, et al. Clinical policy: critical issues in the diagnosis and management of the adult psychiatric patient in the emergency department. Ann Emerg Med. 2006;47(1):79-99.
7. Wilson MP, Nordstrom K, Anderson EL, et al. American Association for Emergency Psychiatry Task Force on Medical Clearance of Adults Part 2: controversies over medical assessment and consensus recommendations. West J Emerg Med. 2017;18(4):640-46.
8. Anderson EL, Nordstrom K, Wilson MP, et al. American Association for Emergency Psychiatry Task Force on Medical Clearance of Adults Part 1: introduction, review, and evidence-based guidelines. West J Emerg Med. 2017;18(2):235-42.
9. Consensus Statement Medical Clearance Protocols for Acute Psychiatric Patients Referred for Inpatient Admission. 2011. Available at: http://www.njha.com/media/33107/ClearanceProtocolsforAcutePsyPatients.pdf. Accessed November 11, 2017.
10. Miller A, Frei SP, Rupp V, et al. Validation of a triage algorithm for psychiatric screening (TAPS) for patients with psychiatric chief complaints. J Am Osteopath Assoc. 2012;112(8):502-8.
11. Shah SS, Fiorito M, McNamara RM. A screening tool to medically clear psychiatric patients in the emergency department. J Emerg Med. 2012;43(5):871-5.
12. Yildiz A, Sasch GS, Turgay . Pharmacological management of agitation in emergency settings. Emerg Med J. 2003;20(4):339-46.
13. Zeller SL, Rhoades RW. Systematic reviews of assessment measures and pharmacologic treatments for agitation. Clin Ther. 2010;32(3):403-25.
14. Wilson MP, Pepper D, Currier GW, et al. The psychopharmacology of agitation: consensus statement of the American Association of Emergency Psychiatry Project BETA Psychopharmacology Workgroup. West J Emerg Med. 2012;13(1):26-34.
15. Gomez S, Dopheide J. Antipsychotic selection for acute agitation and time to repeat use in a psychiatric emergency department. J Psychiatr Prac. 2016;22(6):450-8.
16. Wilson MP, Brennan JJ, Modesti L, et al. Lengths of stay for involuntarily held psychiatric patients in the ED are affected by both patient characteristics and medication use. Am J Emerg Med. 2015;33(4):527-30.
17. MacDonald K, Wilson M, Minassian A, et al. A naturalistic study of intramuscular haloperidol versus intramuscular olanzapine for the management of acute agitation. J Clin Psychopharmacol. 2012;32(3):317-22.
18. McGorry PD, Yung AR, Phillips LJ, et al. Randomized controlled trial of interventions designed to reduce the risk of progression to first episode psychosis in a clinical sample with subthreshold symptoms. Arch Gen Psychiatry. 2002;59(10):921-8.
19. Kane JM, Robinson DG, Schooler NR, et al. Comprehensive versus usual community care for first-episode psychosis: 2-year outcomes from the NIMH RAISE early treatment program. Am J Psychiatry. 2016;173(4):362-72.
20. Byrne P. Managing the acute psychotic episode. BMJ. 2007;334(7595):686-95.
21. National Program Executive Summary Report – Early Detection and Intervention for the Prevention of Psychosis in Adolescents and Young Adults. 2014. Available at: http://www.rwjf.org/content/dam/farm/reports/program_results_reports/2014/rwjf408484/subassets/rwjf408484_1. Accessed April 7, 2017.
22. Diamond RJ. Engaging the crisis patient concerning medication. Emergency Psychiatry: Principles and Practice. 2008:67-78.
23. Beauford JE, McNiel DE, Binder RL. Utility of the initial therapeutic alliance in evaluating psychiatric patients' risk of violence. Am J Psychiatry. 1997;154(9):1272-6.
24. Bessaha ML, Shumway M, Edmondson-Smith M, et al. Predictors of hospital length of stay and cost of stay in a national sample of adult patients with psychotic disorder. Psychiatr Serv. 2017;68(6):559-65.
25. Nolan JM, Fee C, Cooper BA, et al. Psychiatric boarding incidence, duration, and associated factors in United States emergency departments. Emerg Nurs. 2015;41(1):57-64.
26. Park JM, Park LT, Siefert CJ, et al. Factors associated with extended length of stay for patients presenting to an urban psychiatric emergency service: a case-control study. J Behav Health Serv Res. 2009;36(3):300-8.
Articles Behavioral Health Current Issue: Volume 20 Issue 2 Review Article
Previous Post Best Practices for Evaluation and Treatment of Agitated Children and Adolescents (BETA) in the Emergency Department: Consensus Statement of the American Association for Emergency Psychiatry
Next Post A Research Agenda for the Assessment and Management of Acute Behavioral Changes in Elderly Emergency Department Patients
Rapid Discharge After Interfacility Transfer for Mild Traumatic Intracranial Hemorrhage: Frequency and Associated Factors
Stewarding Recovery from the Opioid Crisis Through Health System Initiatives
Addressing Social Determinants of Health from the Emergency Department through Social Emergency Medicine


text:
Affiliation Form
You are at:Home»Article»Rohingyas: victims of ethnic cleansing
Rohingyas: victims of ethnic cleansing
By admin on	 March 12, 2017 Article
By Harun Yahya
Myanmar, undoubtedly the country with the worst record of human rights in Southeast Asia, witnessed a development that aroused excitement and hope across the entire world about a year ago. After half a century of military junta government, the National League for Democracy (NLD) won the November 2015 general elections. One of the election promises of that party's leader and Nobel Peace Prize winner Aung San Suu Kyi was "real change". However, the year that has passed since clearly showed that nothing has ever changed for the Rohingya Muslims: They are still the helpless victims of a horrific ethnic cleansing.
The project of purging Rakhine state of the Rohingyas continues at full steam. Hundreds of innocent Rohingyas have been massacred, hundreds of women have been raped, hundreds have lost their lives while trying to escape persecution and thousands of Muslim houses have been plundered and burned in the ongoing operations of Myanmar's army and security forces since October.
In images published in the press and the media, the innocence and helplessness of the Rohingya people are written all over their faces; the miserable state they are in that is fraught with hunger, poverty and destitution is extremely shocking. In the words of an expert from the United Nations, they are "probably the most friendless people in the world".
Read more at: Burma Times
Rohingya refugee and Bangladeshi women weave a brighter future
A Major Trade Union, a anti-racism Group and British Rohingyas attack Leeds United and its owners
British teachers moved to tears by the Rohingya story
Tweets by @@BritishRohingya
Resume Template Instant Download
Copyright © 2017. Design

redpajama_set_name: RedPajamaCommonCrawl


Westminster, Maryland, U.S.A.: Random House Inc, 1988. First Edition. Hard cover binding, 360 pp. The private and public life of the musical and theatrical genius. Previous owner name inside. Near fine in fine dustjacket.. First Edition. Hard Cover. Near Fine/Fine. 8vo - over 7¾" - 9¾" tall.


text:
January 14, 2021 January 14, 2021 by Rachel Brown Hackney, Editor & Publisher
Board members express hope that extra estimated $5.5 million to $6 million expense can be halved
This rendering shows the District 12 Medical Examiner facility with a second level over the administrative offices portion of the structure. Image courtesy Sarasota County
In early December 2020, the Sarasota County commissioners unanimously authorized staff to continue negotiations with the ownership group that has proposed constructing a new county facility for the District 12 Medical Examiner's Office on about 4.4 acres located at 4480 Fruitville Road in Sarasota, near Interstate 75.
During that discussion, Commissioners Michael Moran and Christian Ziegler called for consideration about the potential addition of a second story to the administrative offices portion of the structure. That would comprise 17,483 square feet.
Moran, who brought up the idea, talked of the county's ability to rent out that space — after it purchases the new building from the ownership group, 4480 Fruitville LLC. Then the county could reap some return on its investment.
Carolyn Eastwood, director of the county's Capital Projects Department, came back to the board members on Jan. 12, explaining the results of the discussions that ensued with the ownership group, which includes John LaCivita, executive vice president of Willis A. Smith Construction in Sarasota.
In short, because of county land-development regulations, Eastwood explained, finishing out a second story over just the administrative offices portion of the building for District 12 Medical Examiner Dr. Russell Vega and his staff would add an estimated $5.5 million to $6 million to the overall expense. That range resulted from the need for 70 more parking spaces — the County Code requires one for every 250 square feet of space; the possible mitigation expense related to removal of grand trees on the property, to allow for the extra height of the structure; and the necessity of including an underground stormwater vault system, as the site is too constrained to enlarge the proposed stormwater pond.
This graphic shows the Medical Examiner's facility site, with the green circles representing grand trees. Image courtesy Sarasota County
After close to 45 minutes of discussion, the commissioners unanimously agreed to approve the proposed lease for the single-story structure, while giving staff authorization to continue negotiations with the ownership group for a second story. In making the motion, Commissioner Moran indicated numerous times his hope that the talks between staff and 4480 Fruitville LLC could end up with about half the estimated expense for the second floor, as the board members agreed, repeatedly, that they wanted just "a cold, dark space," as Moran described it. That phrase referenced a document included in the backup agenda materials that staff had provided the commissioners about the proposed second level.
Moreover, LaCivita told the board members, "I don't think adding the second floor is going to delay the project." Constructing the morgue, he added, is going to be the more intensive part of the undertaking.
Regardless of how the second-level space is used, "The minute we do a ribbon-cutting on this property," Moran said, he would be willing to bet the price of a cup of coffee at Starbucks that the board would get a request from staff regarding the need for 20,000 more square feet "for something. … Mark my words."
The commissioners said they hoped that, through the county's site and development process, the requirement for the 70 additional parking spaces would prove moot. "I think there are uses that may not require as much parking," Commissioner Ron Cutsinger noted. One of those uses might be storage space.
This is part of the document provided in the board packet that describes how the second floor could be constructed as a shell. Image courtesy Sarasota County
Having just been elected in November 2020, Cutsinger talked of tours of county offices that he took in late 2020, as he learns about county operations. "There's boxes stacked everywhere."
County Administrator Jonathan Lewis also noted the nature of site and development discussions, during which staff ultimately could agree to what Lewis referred to as an "alternate parking plan," tailored for specific use of a new building that ends up being a modification of the developer's original vision for the structure. "That's not abnormal," he added of such situations.
Moran acknowledged, "I'm throwing horseshoes and hand grenades here" when he proposed the board would pay much less than the $5.5 million to $6 million for the construction of a shell of a second floor.
He did stress the commission's appreciation to LaCivita and LaCivita's colleagues in the 4480 Fruitville team for the public-private partnership that had led to the Jan. 12 discussion.
Deputy County Administrator and Chief Financial Management Officer Steve Botelho explained that if the commission that day approved the lease for a single-story structure, staff would work on all the details for that addition and come back with a proposed amendment to the agreement.
In the meantime, Botelho said, the design of the facility could proceed.
Eastwood also had noted that if the lease were approved on Jan. 12, 4480 Fruitville LLC anticipated the structure would be ready for Dr. Vega and his staff in January 2023.
This schematic shows plans for use of space in the first level of the administrative offices section. Image courtesy Sarasota County
"Staff has a huge job ahead of 'em," Chair Alan Maio said just before the vote, noting that the county's Development Review Committee (DRC), which will review all the site and development plans and sign off on them before the building gets underway, "is an essential part of what our government does here."
The DRC members represent all the county departments and divisions that are involved in land development.
Based on his career as a principal for many years with the Kimley-Horn consulting firm, Maio indicated that working with the DRC to finalize details of a project could be intense. The commissioners, he said, want the Capital Projects and county administrative staff "to get really entrepreneurial" in those DRC discussions.
Commissioner Christian Ziegler seconded Moran's motion to proceed with the initial lease.
Commissioner Nancy Detert was absent from the meeting, Maio said, because she was "under the weather."
Diving into the financial details
Already, Eastwood of Capital Projects reminded the board members on Jan. 12, the negotiations with 4480 Fruitville LLC had put the total cost of the new Medical Examiner's Office facility right at the $23.5 million cap the Sarasota County Charter imposes on any solitary county project without the need for voter approval, during a referendum, to raise that amount.
A graphic shows the location of the 4480 Fruitville Road property, near I-75. Image courtesy Sarasota County, via Google Maps
Chair Maio said the extra construction funds could come out of impact fee revenue, for example, or even out of the county's General Fund, which comprises property tax and gas tax revenue, for examples.
Further, Eastwood said, with a second story, another $497,000 would be added to the county's annual lease payment to 4480 Fruitville LLC. The lease, she noted, had been expected to run from $4.4 million to $4.9 million per year, based on facets of the final structure. The lease would have an annual escalator of 3%, or the Consumer Price Index value, whichever was higher, she noted.
With property tax payments, insurance and operating costs, Eastwood in December 2020 put the total annual expense for the county as a tenant between $5 million and $5.6 million. The lease would be for 30 years, but 4480 Fruitville LLC would allow the county to purchase the property and building for $19.1 million after a minimum of three years; that time frame was proposed because of lending considerations with which the limited liability company will have to contend.
Categories Construction, Finance, Health & Medicine, Real Estate, Sarasota County Tags Tinypass Leave a comment


Overcome the challenges of Managing IT in a hybrid cloud world.
The IT world is increasingly hybrid, with cloud-based computing making major advances on many areas of on-premise computing. This has opened new opportunities for applications and services that would never have been possible before. Nonetheless, the monolithic "datacenter" of the past, with heavy concentrations of the products of a single vendor in place, has not been completely replaced by the Cloud. Instead, it has given way to a hybrid environment that includes physical datacenters and clouds, decades-old legacy systems, licensed and open-source applications and infrastructure components, and cloud-based services from many different vendors.
Savision's newest whitepaper outlines a framework for approaching IT management that combines excellence in IT Operations with a strong focus on business impact of IT systems.
– What are the different forces that are driving adoption and deployment of hybrid cloud architectures?
– What are the major trends that make today the most innovative environment ever for business, and also the most challenging IT management environment?
– What is the current state of hybrid cloud management offerings?


From the creator of Rapstronaut : Space Journey Game. Have you ever heard of Indonesian food like bakso, martabak, ayam goreng, and nasi bali? Coming to join them and download Warung Chain: Go Food Express directly! Various foods from warteg, martabak, to bakso malang and tahu bulat ready to be cooked and served. Cook and serve delicious foods ranging from snacks, breakfast, lunch, to dinner. With Warung chain: Go food express! You will play as a warung small street food in Indonesian owner trying to share happiness through good delicious food. Warung Chain: Go Food Express is the property and trademark from the developer Touchten. Experience local kindness and play through various popular tourist destinations spots in Indonesia, Encounter diverse customers and obtain knowledge about delicious Indonesian foods and culture. You will play as a warung small street food in Indonesian owner trying to share happiness through good delicious food.
One thing for sure, Customer satisfactions guaranteed!! Warung Chain: Go Food Express is a fast paced cooking game from the makers of Ramen Chain. Various foods from warteg, martabak, to bakso malang and tahu bulat ready to be cooked and served. Warung Chain: Go Food Express is a fast paced cooking game from the makers of Ramen Chain You will play as a warung small street food in Indonesian owner trying to share happiness through good delicious food. Fast reflex, quick thinking and an appetite for famous Indonesian foods is what it takes to play this game! Now you may skilled it on this recreation. Over 58840 users rating a average 4. Its 1,253 downloads rank Warung Chain in the position number 5429 within its category and 136527 of all Android apps. From the creator of Rapstronaut : Space Journey Game.
Experience local kindness and play through various popular tourist destinations spots in Indonesia, Encounter diverse customers and obtain knowledge about delicious Indonesian foods and culture. Cons: 10 characters minimum Count: 0 of 1,000 characters 5. Warung Chain: Go Food Express is a fast paced cooking game from the makers of Ramen Chain. Experience local kindness and play through various popular tourist destinations spots in Indonesia, Encounter diverse customers and obtain knowledge about delicious Indonesian foods and culture. Various foods from warteg, martabak, to bakso malang and tahu bulat ready to be cooked and served.
Play Warung chain: Go food express! From the creator of Rapstronaut : Space Journey Game. Now you can experienced it in this game. Various foods from warteg, martabak, to bakso malang and tahu bulat ready to be cooked and served. Have you ever heard of Indonesian food like bakso, martabak, ayam goreng, and nasi bali? Don't forget to share your delicious meals with your friends on Facebook! Warung Chain: Go Food Express is a fast paced cooking game from the makers of Ramen Chain. Now you can experienced it in this game. And you can have this Warung Chain file for free. Don't forget to share your delicious meals with your friends on Facebook! You can download Warung Chain: Go Food Express 1.
Now do you have what it takes to become a foodpreneur? Hurry, there many be a long queue of hungry customers at your counter. Loads of hungry customers craving for express service in front of your shop, make them happy! Pros: 10 characters minimum Count: 0 of 1,000 characters 4. From the creator of Rapstronaut : Area Journey Recreation. More than 5000000 is playing Warung Chain: Go Food Express right now. Don't forget to share your delicious meals with your friends on Facebook! Warung Chain: Go Food Express is a fast paced cooking game from the makers of Ramen ChainYou will play as a warung small street food in Indonesian owner trying to share happiness through good delicious food. Fast reflex, quick thinking and an appetite for famous Indonesian foods is what it takes to play this game! Now do you have what it takes to become a foodpreneur? Now do you have what it takes to become a foodpreneur? One thing for sure, Customer satisfactions guaranteed!! Cook and serve delicious foods ranging from snacks, breakfast, lunch, to dinner. Now do you have what it takes to become a foodpreneur? Fast reflex, quick thinking and an appetite for famous Indonesian foods is what it takes to play this game! Warung Chain: Go Food Express v1.
Warung Chain: Go Food Express by Touchten Pte. Ltd.
Warung Chain: Go Food Express is a free Arcade game. Experience local kindness and play through various popular tourist destinations spots in Indonesia, Encounter diverse customers and obtain knowledge about delicious Indonesian foods and culture. Fast reflex, quick thinking and an appetite for famous Indonesian foods is what it takes to play this game! Cook and serve delicious foods ranging from snacks, breakfast, lunch, to dinner. Run XePlayer Android Emulator and login Google Play Store. External storage access is used for caching video purpose. Have you ever heard of Indonesian food like bakso, martabak, ayam goreng, and nasi bali? Loads of hungry customers craving for express service in front of your shop, make them happy! More than 5000000 is playing Warung Chain: Go Food Express right now.
Cook and serve delicious foods ranging from snacks, breakfast, lunch, to dinner. Now do you have what it takes to become a foodpreneur? Say no to Google Play and payments for full versions. Cook and serve delicious foods ranging from snacks, breakfast, lunch, to dinner. Loads of hungry customers craving for express service in front of your shop, make them happy! You'll play for hours and just won't be able to tear yourself away from your gadget. You will play as a warung small street food in Indonesian owner trying to share. Loads of hungry customers craving for express service in front of your shop, make them happy! Here I will share the modded or modified version so you can enjoy premium features like Unlimited Money, Gold, Lives.
Loads of hungry customers craving for express service in front of your shop, make them happy! When you recieve an order you will try to make your best and try to feel your customers happy. Download and Install XePlayer Android Emulator. The latest version of Warung Chain: Go Food Express is 1. We provide Warung Chain: Go Food Express 1. Warung chain: Go food express! Various foods from warteg, martabak, to bakso malang and tahu bulat ready to be cooked and served. Summary: optional Count: 0 of 1,500 characters The posting of advertisements, profanity, or personal attacks is prohibited. You will certainly enjoy its fascinating gameplay.
. Experience local kindness and play through various popular tourist destinations spots in Indonesia, Encounter diverse customers and obtain knowledge about delicious Indonesian foods and culture. Cook and serve delicious foods ranging from snacks, breakfast, lunch, to dinner. Send a friend a it. Name Of Game : Warung Chain: Go Food Express Category : Adventure Games Publish Date : 01.


I got this e-mail and so this project happened. "My name is Katie and I work at NPR in Washington, DC. We love your illustration style and were wondering if you would be willing to do a piece for our annual NPR wall calendar for 2011. "


BROOKLYN, N.Y. — It's still too early to know if the North Carolina basketball team has its mojo back after stumbling to a pair of losses to end the regular season.
But Tuesday's ACC Tournament opener against Syracuse was a good start.
Chided by veteran starters Joel Berry and Kenny Williams for being timid in a loss at arch-rival Duke last Saturday, the Tar Heels were in attack mode from the jump. They consistently pounded the ball inside against the Orange's zone and played some of their best defense all year on the way to a liberating 78-59 victory at Barclays Center.
The win sets up a quarterfinal matchup Thursday against Miami, the team that spoiled UNC's Senior Night celebration with a game-winning halfcourt buzzer-beater just over a week ago.
Wearing warmup shirts with the name "Woody" inscribed on the back as a tribute to long-time play-by-play announcer Woody Durham — who died in the early hours of Wednesday morning — the Tar Heels (23-9) looked like the team that reeled off six straight wins in late February rather than the one that dropped consecutive games to Miami and Duke to end the regular season.
They shared the ball, recording 20 assists on 25 baskets, they took the ball right at Syracuse's big men and got both in foul trouble, they pounded the offensive boards to the tune of 20 second-chance points, and they made big shots when they needed them.
Berry even got back on track by making two second half 3-pointers, ending a streak of 11 straight misses dating back to the Duke game.
The duo of Theo Pinson and Luke Maye were especially effective around the rim, with Pinson contributing 16 points, 11 rebounds and six assists while Maye barely missed matching his teammate's double-double with 13 points and nine boards.
Even more important, their ability to get the ball into the middle of the Syracuse zone helped create 10 fouls on Orange big men Paschal Chukwu and Bourama Sidibe — both of whom fouled out.
While UNC used a balanced attack to begin pulling away midway through the first half, Syracuse (20-13) relied almost exclusively on the contributions of just three players. Oshae Brissett, Tyus Battle and Frank Howard combined for 47 of their team's 59 points.
Although they were able to keep the Orange within hailing distance by combining for seven first-half 3-pointers, they had a much more difficult time just getting shots up over a final 20 minutes in which Syracuse shot just 28.6 percent.
The Orange also committed five shot clock violations in the game. The 59 points were the fewest allowed by UNC this season.
"I thought defensively it was one of our better games of the year, if not the best game of the year," Williams said.
After leading by 10 at halftime, UNC broke the game open by starting the second half on a 17-6 run that extended their cushion to 21 at 56-35 with 12:57 remaining.
Then the Tar Heels suddenly lost their aggressiveness.
The Orange took advantage of a five-minute scoring drought to cut the lead to just nine with 4:17 remaining. But just as visions of a second straight 21-point comeback began dancing in the heads of the Syracuse faithful in the crowd — as Notre Dame did to Virginia Tech in the previous game — UNC came to life again.
Two full-court passes that resulted in Kenny Williams layups highlighted a game-ending 12-2 spurt that allowed the Tar Heels to put the game away.
Though not fast enough for their coach's liking.
Instead, he'll have to settle for a win that helped his team get back on the right track again.


text:
Search GoMapper
Home Europe France Poitou-Charentes Vienne Location
Where is Vienne in Poitou-Charentes, France Located?
Vienne is a city/town with a medium population in the state/region of Poitou-Charentes, France which is located in the continent/region of Europe.
Cities, towns and places near Vienne include Dissay, Saint-Georges-lès-Baillargeaux, Saint-Cyr and Montamisé.
The closest major cities include Deux-Sevres, Eure-et-Loir, Indre-et-Loire and Bretagne.
Vienne Data
Latitude 46.6695417 / 46° 40' 10.3506"
Longitude 0.4772865 / 0° 28' 38.2326"
State / Region Poitou-Charentes
Continent Europe
Population Size Medium
List of cities near Vienne
What is the closest airport to Vienne?
List of biggest cities in France
List of countries near France
List of international airports in France
List of airports in France
List of biggest cities in Poitou-Charentes
List of international airports in Poitou-Charentes
List of airports in Poitou-Charentes
GoMapper


text:
\section{Introduction}
Given $m,n\in \mathbb{N}$. When $m\leq n$, let $[m,n]=\{m,m+1,\ldots,n\}$. As usual, let $[n]=\{1,2,\ldots,n\}$.
The cardinality of a set $A$ will be
denoted by $\#A$. Let $\mathcal{S}_n$ be the set of all permutations of $[n]$ and
let $\pi=\pi(1)\pi(2)\cdots\pi(n)\in\mathcal{S}_n$.
Let ${\mathcal S}^B_n$ be the hyperoctahedral group of rank $n$. Elements of ${\mathcal S}^B_n$ are signed permutations $\pi$ of the set $\pm[n]$ such that $\pi(-i)=-\pi(i)$ for all $i$, where $\pm[n]=\{\pm1,\pm2,\ldots,\pm n\}$.
Let $\mathcal{S}^D_n$ denote the group of even signed permutations, which is a Coxeter group of type $D$ of rank $n$.
The group $\mathcal{S}^D_n$ is the subgroup of ${\mathcal S}^B_n$ consisting of signed permutations with an even number of negative entries among $\pi(1),\pi(2)\ldots,\pi(n)$.
Define
\begin{equation*}
\begin{split}
{\rm des\,}_A(\pi)&:=\#\{i\in [n-1]:~\pi(i)>\pi({i+1})\},\\
{\rm des\,}_B(\pi)&:=\#\{i\in [0,n-1]:~\pi(i)>\pi({i+1}),~\pi(0)=0\},\\
{\rm des\,}_D(\pi)&:=\#\{i\in[0,n-1]:~\pi(i)>\pi({i+1}),~\pi(0)=-\pi(2)\}.
\end{split}
\end{equation*}
The types $A,B$ and $D$ Eulerian polynomials are
respectively defined by
\begin{equation*}
\begin{split}
A_n(x)&=\sum_{\pi\in\mathcal{S}_n}x^{{\rm des\,}_A(\pi)},~
B_n(x)=\sum_{\pi\in{\mathcal S}^B_n}x^{{\rm des\,}_B(\pi)},~
D_n(x)=\sum_{\pi\in \mathcal{S}^D_n}x^{{\rm des\,}_D(\pi)}.
\end{split}
\end{equation*}
There is a close connection among the three types of Eulerian polynomials~\cite[Lemma 9.1]{Stembridge94}:
\begin{equation}\label{Dnx-BnxAnx}
D_n(x)=B_n(x)-n2^{n-1}xA_{n-1}(x) \quad {\text for}\quad n\geq 2.
\end{equation}
Subsequently, Brenti~\cite[Theorem~4.7]{Brenti94} obtained a $q$-analogue of~\eqref{Dnx-BnxAnx}. There are several recurrences for $D_n(x)$ (see~\cite{Chow03,Hyatt16}).
The polynomials $A_n(x),B_n(x)$ and $D_n(x)$ have several common properties, including unimodality, real-rootedness and $\gamma$-positivity
(see~\cite{Athanasiadis17,Hyatt16,Liu07,Petersen07,Savage15,Yang}).
This paper is motivated by the recent work of Hyatt~\cite{Hyatt16} and Hwang, Chern and Duh~\cite{Hwang20}.

Let ${\mathcal S}^B_n=B_n^+\cup B_n^-$ and $\mathcal{S}^D_n=D_n^+\cup D_n^-$, where
$$B_n^+=\{\pi\in{\mathcal S}^B_n: \pi(n)>0\},~B_n^-=\{\pi\in{\mathcal S}^B_n:~\pi(n)<0\},$$
$$D_n^+=\{\pi\in\mathcal{S}^D_n: \pi(n)>0\},~D_n^-=\{\pi\in\mathcal{S}^D_n:~\pi(n)<0\}.$$
Define $$P_n(x)=\sum_{k=0}^{n-1}\binom{n}{k}B_k(x)(x-1)^{n-k-1},~Q_n(x)=\sum_{k=0}^{n-1}\binom{n}{k}D_k(x)(x-1)^{n-k-1}.$$
Given a polynomial $f(x)$ of degree $n$. Let $\widetilde{f}(x)=x^nf(1/x)$.
In~\cite{Hyatt16}, Hyatt found that
\begin{equation}\label{Bnx}
B_n(x)=P_n(x)+x^nP_n(1/x)=P_n(x)+x\widetilde{P}_n(x)~\text{for $n\geq 1$},
\end{equation}
\begin{equation*}\label{Dnx}
D_n(x)=Q_n(x)+x^nQ_n(1/x)=Q_n(x)+x\widetilde{Q}_n(x)~\text{for $n\geq 2$},
\end{equation*}
where
\begin{equation*}
P_n(x)=\sum_{\pi\in B_n^+}x^{{\rm des\,}_B(\pi)},~x^nP_n(1/x)=\sum_{\pi\in B_n^-}x^{{\rm des\,}_B(\pi)},
\end{equation*}
\begin{equation*}
Q_n(x)=\sum_{\pi\in D_n^+}x^{{\rm des\,}_D(\pi)},~x^nQ_n(1/x)=\sum_{\pi\in D_n^-}x^{{\rm des\,}_D(\pi)}.
\end{equation*}
Motivated by~\eqref{Bnx}, in the following we shall present a more natural decomposition of $B_n(x)$.

Following~\cite{Adin2001}, the {\it flag descent number} of $\pi\in{\mathcal S}^B_n$
is defined by
\begin{equation*}
{\rm fdes\,}(\pi):=\begin{cases}
2{\rm des\,}_A(\pi)+1,& \text{if $\pi(1)<0$};\\
2{\rm des\,}_A(\pi), & \text{otherwise}.
\end{cases}
\end{equation*}
Clearly, ${\rm fdes\,}(\pi)={\rm des\,}_A(\pi)+{\rm des\,}_B(\pi)$. The {\it flag descent polynomial} is defined by
$$C_n(x)=\sum_{\pi\in {\mathcal S}^B_n}x^{{\rm fdes\,}(\pi)}.$$
It follows from~\cite[Theorem~4.4]{Adin2001} that
\begin{equation}\label{fdesAnx}
C_n(x)=(1+x)^nA_n(x),
\end{equation}
and so $C_n(x)$ is symmetric and unimodal. Let ${\rm neg\,}(\pi):=\#\{i\in [n]:~\pi(i)<0\}$.
Consider the $q$-flag descent polynomials
$$C_n(x,q)=\sum_{\pi\in {\mathcal S}^B_n}x^{{\rm fdes\,}(\pi)}q^{{\rm neg\,}(\pi)}.$$
Set ${\mathcal S}^B_n=C_n^+\cup C_n^-$, where $C_n^+=\{\pi\in{\mathcal S}^B_n:~\pi(1)>0\}$ and $C_n^-=\{\pi\in {\mathcal S}^B_n:~\pi(1)<0\}$.
Define
$$C_n^E(x,q)=\sum_{\pi\in C_n^+}x^{{\rm des\,}_A(\pi)}q^{{\rm neg\,}(\pi)},~C_n^O(x,q)=\sum_{\pi\in C_n^-}x^{{\rm des\,}_A(\pi)}q^{{\rm neg\,}(\pi)}.$$
Then we have
\begin{equation}\label{SnxqEO}
C_n(x,q)=C_n^E(x^2,q)+xC_n^O(x^2,q).
\end{equation}

Consider the {\it type $B$ $q$-Eulerian polynomials}
$$B_n(x,q)=\sum_{\pi\in {\mathcal S}^B_n}x^{{\rm des\,}_B(\pi)}q^{{\rm neg\,}(\pi)}.$$
The polynomials $B_n(x,q)$ satisfy the recurrence
$$B_{n+1}(x,q)=((1+q)nx+qx+1)B_n(x,q)+(1+q)x(1-x)B_n'(x,q),$$
with $B_0(x,q)=1$ (see~\cite[Theorem~3.4]{Brenti94}).
Note that ${\rm des\,}_B(\pi)={\rm des\,}_A(\pi)$ for $\pi\in C_n^+$ and ${\rm des\,}_B(\pi)={\rm des\,}_A(\pi)+1$ for $\pi\in C_n^-$.
Hence
\begin{equation*}
B_n(x,q)=C_n^E(x,q)+xC_n^O(x,q).
\end{equation*}
Set $C_n^E(x,1)=C_n^E(x)$ and $C_n^O(x,1)=C_n^O(x)$. By~\eqref{fdesAnx}, the polynomial $C_n(x)$ is symmetric. Thus $C_n^E(x)=x^{n-1}C_n^O(1/x)$ for $n\geq 1$.
Note that $B_n(x,1)=B_n(x)$ and $C_n(x,1)=C_n(x)$. We can now conclude the following result from the discussion above.
\begin{proposition}\label{SnxBnx-prop}
For $n\geq 1$, we have
\begin{equation}\label{SnxBnx-decom}
C_n(x)=C_n^E(x^2)+xC_n^O(x^2),~B_n(x)=C_n^E(x)+xC_n^O(x).
\end{equation}
\end{proposition}

Comparing~\eqref{Bnx} with~\eqref{SnxBnx-decom}, we get the following result, and we give a proof of it for completeness.
\begin{proposition}
We have $C_n^E(x)=P_n(x)$ and $C_n^O(x)=\widetilde{P}_n(x)$.
\end{proposition}
\begin{proof}
Define
\begin{align*}
{^+B_n^+}&=\{\pi\in B_n: \pi(1)>0,~ \pi(n)>0\},\\
{^+B_n^-}&=\{\pi\in B_n: \pi(1)>0,~ \pi(n)<0\},\\
{^-B_n^+}&=\{\pi\in B_n: \pi(1)<0,~ \pi(n)>0\},\\
{^-B_n^-}&=\{\pi\in B_n: \pi(1)<0,~ \pi(n)<0\}.
\end{align*}
Note that $C_n^+={^+B_n^+}\cup {^+B_n^-}$ and $B_n^+={^+B_n^+}\cup {^-B_n^+}$.
A bijection $\Phi$ from $C_n^+$ to $B_n^+$ is given as follows:
\begin{itemize}
  \item [$(i)$] If $\pi\in {^+B_n^+}$, then let $\Phi(\pi)=\pi$;
  \item  [$(ii)$] For $\pi\in {^+B_n^-}$, let $k$ be the smallest index of $\pi$ such that $\pi(k)>0$ and $\pi(k+1)<0$.
  Then we define $\Phi(\pi)=\pi(k+1)\cdots \pi(n)\pi(1)\cdots\pi(k)$.
\end{itemize}
Note that ${\rm des\,}_B\left(\Phi(\pi)\right)={\rm des\,}_B(\pi)$. Hence
\begin{equation*}
\sum_{\pi\in C_n^+}x^{{\rm des\,}_B(\pi)}=\sum_{\pi\in B_n^+}x^{{\rm des\,}_B(\pi)},
\end{equation*}
Along the same lines, it is easy to show that
$\sum_{\pi\in C_n^-}x^{{\rm des\,}_B(\pi)}=\sum_{\pi\in B_n^-}x^{{\rm des\,}_B(\pi)}$.
\end{proof}

Motivated by Proposition~\ref{SnxBnx-prop}, in the following we shall establish a duality relation between Eulerian recurrences and Eulerian recurrence systems.
It is well known that the Eulerian polynomials $A_n(x)$ and $B_n(x)$ satisfy the following recurrence relations:
\begin{equation*}
\begin{split}
A_{n}(x)&=(nx+1-x)A_{n-1}(x)+x(1-x)A_{n-1}'(x),\\
B_{n}(x)&=(2nx+1-x)B_{n-1}(x)+2x(1-x)B_{n-1}'(x),
\end{split}
\end{equation*}
with the initial conditions $A_0(x)=B_0(x)=1$ (see~\cite{Brenti94,Zhuang17}).
In recent years, there has been much work on the generalizations of Eulerian recurrences, see~\cite{Barbero14,Hwang20,Zhu} and references therein.
In~\cite{Barbero14}, Barbero, Salas and Villase\~{n}or systematically studied and classified the partial differential equations that are satisfied by
the generating function
$$f(x,y)=\sum_{n,k\geq 0}\eulerian{n}{k}x^k\frac{y^n}{n!},$$
where the numbers $\eulerian{n}{k}$ satisfy the recurrence relation
$$\eulerian{n}{k}=(\alpha n+\beta k+\gamma)\eulerian{n-1}{k-1}+(\alpha' n+\beta' k+\gamma')\eulerian{n-1}{k},$$
with $\eulerian{0}{0}=1$ and $\eulerian{n}{k}=0$ when $n<0$ or $k<0$.
Very recently, Hwang, Chern and Duh~\cite{Hwang20} considered the general Eulerian recurrence:
\begin{equation}\label{Eulerian02}
P_{n}(x)=(\alpha(x)n+\gamma(x))P_{n-1}(x)+\beta(x)(1-x)P_{n-1}'(x),
\end{equation}
with $P_0(x),\alpha(x),\beta(x)$ and $\gamma(v)$ are given functions of $x$ (they are often polynomials). They studied the limiting distribution of the coefficients of $P_n(x)$ for large $n$ when the coefficients are nonnegative. In particular, Hwang, Chern and Duh~\cite[Section~9.3]{Hwang20} discussed the limiting distribution of the coefficients polynomials that satisfy Eulerian recurrence systems.

\begin{definition}\label{def01}
Let $\{E_n(x)\}_{n\geq 0}$ and $\{O_n(x)\}_{n\geq 0}$ be two sequences of polynomials.
We say that the ordered pair of polynomials $(E_n(x),O_n(x))$ is a Eulerian pair if $\deg E_n(x)\geq \deg O_n(x)$ and
the polynomials $E_n(x)$ and $O_n(x)$ satisfy the Eulerian recurrence system:
\begin{equation}\label{EulerianPair}
\left\{
  \begin{array}{l}
    E_{n+1}(x)=p_n(x)E_n(x)+q_n(x)E_n'(x)+r_n(x)O_n(x), \\
    O_{n+1}(x)=u_n(x)O_n(x)+v_n(x)O_n'(x)+w_n(x)E_n(x),
  \end{array}
\right.
\end{equation}
where $E_0(x),O_0(x),p_n(x),q_n(x),r_n(x),u_n(x),v_n(x),w_n(x)$ are given polynomials of low degrees.
\end{definition}

Let $f(x)=\sum_{i=0}^nf_ix^i$. Throughout this paper, we always let
\begin{align*}
f^E(x)&=\sum_{k=0}^{\lrf{\frac{n}{2}}}f_{2k}x^{k},~f^O(x)=\sum_{k=0}^{\lrf{\frac{n-1}{2}}}f_{2k+1}x^{k};\\
f^e(x)&=\sum_{k=0}^{\lrf{\frac{n}{2}}}f_{2k}x^{2k},~f^o(x)=\sum_{k=0}^{\lrf{\frac{n-1}{2}}}f_{2k+1}x^{2k+1}.
\end{align*}
Hence $f(x)=f^E(x^2)+xf^O(x^2)=f^e(x)+f^o(x)$.
As an extension of~\eqref{Eulerian02}, we define a sequence of polynomials $F_n(x)$ by using the following general Eulerian recurrence:
\begin{equation}\label{Eulerian03}
F_{n+1}(x)=\alpha_n(x)F_{n}(x)+\beta_n(x)F_{n}'(x),
\end{equation}
where $F_0(x),\alpha_n(x)$ and $\beta_n(x)$ are given polynomials of low degrees.
We can now present the first main result of this paper.
\begin{theorem}\label{mathm01}
Let $\langle E_n(x),O_n(x)\rangle$ be a Eulerian pair that satisfies the Eulerian recurrence syetem~\eqref{EulerianPair},
and let $F_n(x)$ be the polynomial defined by the recurrence~\eqref{Eulerian03}.
Then the polynomial $F_n(x)$ has the expression $F_n(x)=E_n(x^2)+xO_n(x^2)$ if and only if the following conditions hold:
\begin{align*}
u_n(x)&=p_n(x)+\frac{1}{2x}q_n(x),~v_n(x)=q_n(x),~w_n(x)=\frac{1}{x}r_n(x),\\
\alpha_n(x)&=p_n(x^2)+\frac{1}{x}r_n(x^2),~\beta_n(x)=\frac{1}{2x}q_n(x^2),~\beta_n^e(x)=0.
\end{align*}
\end{theorem}
\begin{proof}
By using $F_n(x)=E_n(x^2)+xO_n(x^2)$, we obtain
\begin{equation*}
F_n'(x)=2xE_n'(x^2)+O_n(x^2)+2x^2O_n'(x^2).
\end{equation*}
Then it follows from~\eqref{Eulerian03} that
\begin{align*}
F_{n+1}(x)&=\alpha_n(x)\left(E_n(x^2)+xO_n(x^2)\right)+\\
&\beta_n(x)\left(2xE_n'(x^2)+O_n(x^2)+2x^2O_n'(x^2)\right).
\end{align*}
Comparing this with the expression
$F_{n+1}(x)=E_{n+1}(x^2)+xO_{n+1}(x^2)$, we obtain
\begin{align*}
E_{n+1}(x^2)&=\alpha_n^e(x)E_n(x^2)+x\alpha_n^o(x)O_n(x^2)+\\
&\beta_n^e(x)\left(O_n(x^2)+2x^2O_n'(x^2)\right)+2x\beta_n^o(x)E_n'(x^2),\\
O_{n+1}(x^2)&=\frac{1}{x}\alpha_n^o(x)E_n(x^2)+\alpha_n^e(x)O_n(x^2)+\\
&\frac{1}{x}\beta_n^o(x)\left(O_n(x^2)+2x^2O_n'(x^2)\right)+2\beta_n^e(x)E_n'(x^2).
\end{align*}
Since~\eqref{EulerianPair} holds, then $\beta_n^e(x)=0$.
Hence
\begin{equation}\label{EulerianPair02}
\left\{
  \begin{array}{l}
E_{n+1}(x^2)=\alpha_n^e(x)E_n(x^2)+2x\beta_n^o(x)E_n'(x^2)+x\alpha_n^o(x)O_n(x^2),\\
O_{n+1}(x^2)=\left(\alpha_n^e(x)+\frac{1}{x}\beta_n^o(x)\right)O_n(x^2)+2x\beta_n^o(x)O_n'(x^2)+\frac{1}{x}\alpha_n^o(x)E_n(x^2).
  \end{array}
\right.
\end{equation}
By comparing~\eqref{EulerianPair} with~\eqref{EulerianPair02}, we immediately get the following relations:
\begin{align*}
p_n(x^2)&=\alpha_n^e(x),~q_n(x^2)=2x\beta_n^o(x),~r_n(x^2)=x\alpha_n^o(x),\\
u_n(x^2)&=\alpha_n^e(x)+\frac{1}{x}\beta_n^o(x),~v_n(x^2)=2x\beta_n^o(x),~w_n(x^2)=\frac{1}{x}\alpha_n^o(x),
\end{align*}
which yield the desired result. This completes the proof.
\end{proof}

Let $F_n(x)$ be the polynomials defined by the recurrence~\eqref{Eulerian03}.
When $\beta_n^e(x)=0$, we can get the recurrence system~\ref{EulerianPair} by using
the following relations:
\begin{equation}\label{dualrelation01}
x\alpha_n(x)=xp_n(x^2)+r_n(x^2),~2x\beta_n(x)=q_n(x^2).
\end{equation}

A polynomial $p\in\R[x]$ is ({\it Hurwitz} or {\it asymptotically})
{\it stable} if every zero of $p$ is in the open left half plane
$\LHP=\{z\colon\re z<0\}$;
$p$ is {\it standard} if its leading coefficient is positive.
Suppose that $p,q\in\R[x]$ both have only real zeros,
that those of $p$ are $\xi_1\leqslant\cdots\leqslant\xi_n$,
and that those of $q$ are $\theta_1\leqslant\cdots\leqslant\theta_m$.
We say that $p$ {\it interlaces} $q$ if $\deg q=1+\deg p$ and the zeros of
$p$ and $q$ satisfy
$$
\theta_1\leqslant\xi_1\leqslant\theta_2\leqslant\cdots\leqslant\xi_n
\leqslant\theta_{n+1}.
$$
We also say that $p$ {\it alternates left of} $q$ if $\deg p=\deg q$
and the zeros of $p$ and $q$ satisfy
$$
\xi_1\leqslant\theta_1\leqslant\xi_2\leqslant\cdots\leqslant\xi_n
\leqslant\theta_n.
$$
We use the notation $p\prec q$ for
either ``$p$ interlaces $q$'' or ``$p$ alternates left of $q$''.
The next theorem is a version of the classical Hermite-Biehler theorem.
\begin{theorem}[{\cite[Theorem~4.1]{Branden11}}]\label{Hermite}
Let $F(x)=F^E(x^2)+xF^O(x^2)\in\R[x]$ be standard.
Then $F(x)$ is stable if and only if both $F^E(x)$ and $F^O(x)$ are standard,
have only nonpositive zeros, and $F^O(x)\prec F^E(x)$.
\end{theorem}

The Hermite-Biehler theorem has been widely used to study the distribution of zeros. In this paper, we consider combinatorial aspects of the expression $F(x)=F^E(x^2)+xF^O(x^2)$, which we call the {\it Hermite-Biehler decomposition} of $F(x)$.
In Section~\ref{Section02}, we collect the Hermite-Biehler decompositions of several enumerative polynomials,
including flag descent polynomials for hyperoctahedral group, flag ascent-plateau polynomials for Stirling permutations, up-down run polynomials for symmetric group and
alternating run polynomials for hyperoctahedral group. As applications of Theorem~\ref{mathm01}, we get some new properties of associated polynomials.
\section{Eulerian recurrence systems and Hermite-Biehler decompositions}\label{Section02}
\subsection{The flag descent polynomials}
\hspace*{\parindent}

It follows from~\cite[Theorem 4.3]{Adin2001} that
the flag descent polynomials $C_n(x)$ satisfy the recurrence
\begin{equation}\label{AdinSnx-recu}
C_{n+1}(x)=(2nx^2+x+1)C_n(x)+x(1-x^2)C_n'(x),
\end{equation}
with $C_0(x)=1,C_1(x)=1+x$ and $C_2(x)=1+3x+3x^2+x^3$.
Set $$\alpha_n(x)=2nx^2+x+1,~\beta_n(x)=x(1-x^2).$$ Note that $\beta_n^e(x)=0$.
It follows from~\eqref{dualrelation01} that
$$x(2nx^2+x+1)=xp_n(x^2)+r_n(x^2),~2x^2(1-x^2)=q_n(x^2).$$
Hence $p_n(x)=2nx+1,~q_n(x)=2x(1-x),~r_n(x)=x$.
By Theorem~\ref{mathm01}, we see that
$$u_n(x)=2nx+1+1-x=(2n-1)x+2,~v_n(x)=2x(1-x),~w_n(x)=1.$$
By Theorem~\ref{mathm01} and Theorem~\ref{Hermite}, we get the second main result of this paper.
\begin{theorem}\label{Snx-thm}
For $n\geq1$, we have
\begin{equation}\label{Snx-recu}
\left\{
  \begin{array}{l}
    C_{n+1}^E(x)=(2nx+1)C_n^E(x)+2x(1-x)\frac{d}{dx}{C_n^E}(x)+xC_n^O(x), \\
    C_{n+1}^O(x)=((2n-1)x+2)C_n^O(x)+2x(1-x)\frac{d}{dx}C_n^O(x)+C_n^E(x),
  \end{array}
\right.
\end{equation}
with the initial conditions $C_1^E(x)=C_1^O(x)=1$. Moreover, both $C_n^E(x)$ and $C_n^O(x)$ have only real negative zeros and $C_n^O(x)\prec C_n^E(x)$.
\end{theorem}

Define
$$C^E(x,z)=1+\sum_{n=1}^\infty C_n^E(x)\frac{z^n}{n!},~C^O(x,z)=\sum_{n=1}^\infty C_n^O(x)\frac{z^n}{n!}.$$
By rewriting~\eqref{Snx-recu} in terms of generating functions, we have
\begin{equation}\label{Snx-recu-GF}
\left\{
  \begin{array}{l}
    (1-2xz)\frac{\partial}{\partial z}C^E(x,z)=C^E(x,z)+2x(1-x)\frac{\partial}{\partial x}C^E(x,z)+xC^O(x,z),\\
      (1-2xz)\frac{\partial}{\partial z}C^O(x,z)=(2-x)C^O(x,z)+2x(1-x)\frac{\partial}{\partial x}C^O(x,z)+C^E(x,z).
  \end{array}
\right.
\end{equation}
It is routine to check that
\begin{equation}\label{SnE-GF}
C^E(x,z)=\frac{x-e^{(x-1)z}}{x-e^{2(x-1)z}},~C^O(x,z)=\frac{1-e^{(x-1)z}}{e^{2(x-1)z}-x}.
\end{equation}

It is well known that the exponential generating functions of $A_n(x)$ and $B_n(x)$ are given as follows (see~\cite[Theorem~3.4]{Brenti94}):
\begin{equation*}\label{An-GF}
A(x,z):=\sum_{n=0}^\infty A_n(x)\frac{z^n}{n!}=\frac{x-1}{x-e^{(x-1)z}}=\frac{(1-x)e^{(1-x)z}}{1-xe^{(1-x)z}},
\end{equation*}
\begin{equation*}\label{An-GF}
B(x,z):=\sum_{n=0}^\infty B_n(x)\frac{z^n}{n!}=\frac{(1-x)e^{(1-x)z}}{1-xe^{2(1-x)z}}.
\end{equation*}

We can now present the third main result of this paper.
\begin{theorem}\label{mainthm03}
We have
$$\frac{A(x,2z)}{A(x,z)}=C^E(x,z),~\frac{B(x,z)}{A(x,z)}=1+xC^O(x,z).$$
\end{theorem}
\begin{proof}
Note that
$$A(x,2z)=\frac{x-1}{x-e^{2(x-1)z}}=\frac{x-e^{(x-1)z}}{x-e^{2(x-1)z}}\frac{x-1}{x-e^{(x-1)z}}=C^E(x,z)A(x,z),$$
which yields the first formula. Since
\begin{equation*}
1+xC^O(x,z)=\frac{1-xe^{(1-x)z}}{1-xe^{2(1-x)z}},
\end{equation*}
it follows that
$$B(x,z)=\frac{(1-x)e^{(1-x)z}}{1-xe^{2(1-x)z}}=\frac{1-xe^{(1-x)z}}{1-xe^{2(1-x)z}}\frac{(1-x)e^{(1-x)z}}{1-xe^{(1-x)z}}=\left(1+xC^O(x,z)\right)A(x,z).$$
This completes the proof.
\end{proof}
An immediate consequence of Theorem~\ref{mainthm03} is the following corollary.
\begin{corollary}
For $n\geq 0$, we have
$$2^nA_n(x)=\sum_{k=0}^n\binom{n}{k}A_k(x)C_{n-k}^E(x),$$
$$B_n(x)=A_n(x)+x\sum_{k=0}^{n-1}\binom{n}{k}A_k(x)C_{n-k}^O(x).$$
\end{corollary}

In~\cite{Bagno04}, Bagno and Garber introduced the definition of flag excedance of signed permutations.
Let $\pi\in{\mathcal S}^B_n$.
The {\it flag excedance number} of $\pi$ is defined by
$${\rm fexc\,}(\pi):=2\#\{i\in [n]:~\pi(i)>i\}+\#\{i\in [n]:~\pi(i)<0\}.$$
Let ${\rm exc\,}_A(\pi)=\#\{i\in [n]:~\pi(i)>i\}$.
Then ${\rm fexc\,}(\pi)=2{\rm exc\,}_A(\pi)+{\rm neg\,}(\pi)$.
It is well known that
\begin{equation}\label{fdesfexc}
\sum_{\pi\in{\mathcal S}^B_n}x^{{\rm fdes\,}(\pi)}=\sum_{\pi\in{\mathcal S}^B_n}x^{{\rm fexc\,}(\pi)},
\end{equation}
which has been extended to colored permutations and affine Weyl groups.
Ordinary and $q$-generalizations of~\eqref{fdesfexc} have been pursued by several authors, see~\cite{Foata11,Mongelli13,Zeng16} for instance.
For example, Mongelli~\cite{Mongelli13} derived some basic properties of the flag excedance polynomials of classical and affine Weyl groups. In particular,
Mongelli~\cite[p.~1221-1222]{Mongelli13} presented a combinatorial proof of~\eqref{fdesfexc}. Recall that $\mathcal{S}^D_n$ is the Coxeter group of type $D$.
According to~\cite[Corollary~5.1]{Mongelli13},
\begin{equation}\label{Dnfexc}
\sum_{\pi\in\mathcal{S}^D_n}x^{{\rm fexc\,}(\pi)}=\frac{1}{2}\left((1+x)^nA_n(x)+(1-x)^nA_n(-x)\right).
\end{equation}

By~\eqref{fdesfexc}, we see that $$C_n(x)=\sum_{\pi\in\mathcal{S}^D_n}x^{{\rm fexc\,}(\pi)}+\sum_{\pi\in{\mathcal S}^B_n\setminus\mathcal{S}^D_n}x^{{\rm fexc\,}(\pi)}.$$
Let $C(x,z):=\sum_{n=0}^\infty C_n(x)\frac{z^n}{n!}$. It follows from~\eqref{fdesAnx} that $C(x,z)=A(x,(1+x)z)$.
By using~\eqref{Dnfexc}, we immediately get
\begin{equation*}
\sum_{n=0}^\infty \sum_{\pi\in\mathcal{S}^D_n}x^{{\rm fexc\,}(\pi)}\frac{z^n}{n!}=\frac{1}{2}\left(C(x,z)+C(-x,z)\right)=C^E(x^2,z).
\end{equation*}
Thus
\begin{equation*}
\sum_{n=0}^\infty \sum_{\pi\in{\mathcal S}^B_n\setminus \mathcal{S}^D_n}x^{{\rm fexc\,}(\pi)}\frac{z^n}{n!}=xC^O(x^2,z).
\end{equation*}
So we get the following result, which gives a connection between Coxeter groups ${\mathcal S}^B_n$ and $\mathcal{S}^D_n$.
\begin{proposition}\label{CnxBnxdes}
Let $C_n^+=\{\pi\in{\mathcal S}^B_n:~\pi(1)>0\}$. Then we have
\begin{equation*}
\sum_{\pi\in\mathcal{S}^D_n}x^{{\rm fexc\,}(\pi)}=\sum_{\pi\in C_n^+}x^{2{\rm des\,}_A(\pi)}.
\end{equation*}
\end{proposition}
A combinatorial proof of Proposition~\ref{CnxBnxdes} would be interesting.
\subsection{Flag ascent-plateau polynomials for Stirling permutations}
\hspace*{\parindent}

Stirling permutations were introduced by Gessel and Stanley~\cite{Gessel78}.
A {\it Stirling permutation} of order $n$ is a permutation of the multiset $\{1,1,2,2,\ldots,n,n\}$ such that
for each $i$, $1\leq i\leq n$, all entries between the two occurrences of $i$ are larger than $i$.
The reader is referred to~\cite{Bona08,Haglund12,Ma17,Ma2001} for the recent study on Stirling permutations.

Denote by $\mathcal{Q}_n$ the set of {\it Stirling permutations} of order $n$.
Let $\sigma=\sigma_1\sigma_2\cdots\sigma_{2n}\in\mathcal{Q}_n$.
The {\it ascent-plateau number}, {\it left ascent-plateau number} and {\it flag ascent-plateau number} of $\sigma$ are respectively defined by
\begin{align*}
{\rm ap\,}(\pi)&=\#\{i\in[2,2n-1]:~\pi(i-1)<\pi(i)=\pi(i+1)\},\\
{\rm lap\,}(\pi)&=\#\{i\in[2n-1]:~\pi(i-1)<\pi(i)=\pi(i+1),~\pi(0)=0\},\\
{\rm fap\,}(\pi)&=\left\{
               \begin{array}{ll}
                 2{\rm ap\,}(\sigma)+1, & \hbox{if $\sigma_1=\sigma_2$;} \\
                 2{\rm ap\,}(\sigma), & \hbox{otherwise.}
               \end{array}
             \right.
\end{align*}
Clearly, ${\rm fap\,}(\sigma)={\rm ap\,}(\sigma)+{\rm lap\,}(\sigma)$.
The {\it flag ascent-plateau polynomials} $L_n(x)$ are defined by $$L_n(x)=\sum_{\sigma\in\mathcal{Q}_n}x^{{\rm fap\,}(\sigma)}.$$
The polynomials $L_n(x)$ satisfy the recurrence relation
\begin{equation}\label{Fnx-recu}
L_{n+1}(x)=(x+2nx^2)L_n(x)+x(1-x^2)L_n'(x),
\end{equation}
with $L_0(x)=1$ (see~\cite[p.~14]{Ma2001}). The first few $L_n(x)$ are
\begin{align*}
L_1(x)&=x,\\
L_2(x)&=x+x^2+x^3,\\
L_3(x)&=x+3x^2+7x^3+3x^4+x^5.
\end{align*}
Set $\mathcal{Q}_n=\mathcal{Q}_n^+\cup\mathcal{Q}_n^-$, where $\mathcal{Q}_n^+=\{\sigma\in\mathcal{Q}_n:~\sigma_1<\sigma_2\}$ and $\mathcal{Q}_n^-=\{\sigma\in\mathcal{Q}_n:~\sigma_1=\sigma_2\}$.
Then
\begin{equation}\label{Lnx-express}
L_n(x)=L_n^E(x^2)+xL_n^O(x^2)=\sum_{\sigma\in\mathcal{Q}_n^+}x^{2{\rm ap\,}(\sigma)}+x\sum_{\sigma\in\mathcal{Q}_n^-}x^{2{\rm ap\,}(\sigma)}.
\end{equation}

From~\eqref{Fnx-recu}, we see that $\alpha_n(x)=x+2nx^2,~\beta_n(x)=x(1-x^2)$ and $\beta_n^e(x)=0$.
Using~\eqref{dualrelation01}, we obtain $x(x+2nx^2)=xp_n(x^2)+r_n(x^2)$ and $2x^2(1-x^2)=q_n(x^2)$.
Hence $$p_n(x)=2nx,~q_n(x)=2x(1-x),~r_n(x)=x,$$
$$u_n(x)=2nx+1-x=(2n-1)x+1,~v_n(x)=2x(1-x),~w_n(x)=1.$$
Therefore, by Theorem~\ref{mathm01}, we can now present the fouth main result of this paper.
\begin{theorem}
For $n\geq1$, we have
\begin{equation}\label{Tnx-recu}
\left\{
  \begin{array}{l}
    L_{n+1}^E(x)=2nxL_n^E(x)+2x(1-x)\frac{d}{dx}{L_n^E}(x)+xL_n^O(x), \\
    L_{n+1}^O(x)=((2n-1)x+1)L_n^O(x)+2x(1-x)\frac{d}{dx}L_n^O(x)+L_n^E(x),
  \end{array}
\right.
\end{equation}
with the initial conditions $L_1^E(x)=0$ and $L_1^O(x)=1$.
\end{theorem}

The {\it ascent-plateau polynomials} and {\it left ascent-plateau polynomials} are defined by
$$M_n(x)=\sum_{\sigma\in\mathcal{Q}_n}x^{{\rm ap\,}(\pi)},~N_n(x)=\sum_{\sigma\in\mathcal{Q}_n}x^{{\rm lap\,}(\pi)}.$$
According to~\cite[Theorem 2,~Theorem 3]{Ma15}, we have
\begin{equation*}\label{Mxt}
M(x,t)=\sum_{n\geq 0}M_n(x)\frac{t^n}{n!}=\sqrt{\frac{x-1}{x-e^{2t(x-1)}}},
\end{equation*}
\begin{equation*}\label{Nxt}
N(x,t)=\sum_{n\geq 0}N_n(x)\frac{t^n}{n!}=\sqrt{\frac{1-x}{1-xe^{2t(1-x)}}}.
\end{equation*}
Clearly, $M_n(x)=x^nN_n\left(\frac{1}{x}\right)$. Moreover, according to~\cite[Proposition~1]{Ma17}, we have
\begin{align*}
2^nxA_n(x)&=\sum_{i=0}^n\binom{n}{i}N_i(x)N_{n-i}(x),\\
B_n(x)&=\sum_{i=0}^n\binom{n}{i}N_i(x)M_{n-i}(x).
\end{align*}
It is well known that $A_n(x)$ and $B_n(x)$ are both unimodal (see~\cite{Athanasiadis17} for instance).
Motivated by the above convolution formulas, it is natural to explore the unimodality of $M_n(x)$ and $N_n(x)$.

Let $f(x)=\sum_{i=0}^nf_ix^i$.
We say that $f(x)$ is {\it unimodal} if there exists an index $0\leq k\leq n$ such that $f_0\leq f_1\leq \cdots\leq f_k\geq f_{k+1}\geq\cdots \geq f_n$.
Such an index $k$ is called a {\it mode} of $f(x)$. In the past decades, it is a fundamental problem to determine the location of modes of a sequence
of combinatorial polynomials, see~\cite{Chen09,Wang2005} for instance.
Following~\cite[Definition~2.9]{Schepers13},
we say that $f(x)$ is {\it alternatingly increasing} if
$$f_0\leq f_n\leq f_1\leq f_{n-1}\leq\cdots f_{\lrf{\frac{n+1}{2}}}.$$
Clearly, alternatingly increasing property is a stronger property than unimodality. Very recently,
there has been much work on the alternatingly increasing property of Ehrhart polynomials (see~\cite{Beck2019,Branden18,Solus19}).
In the rest part of this subsection, we shall show that
both $M_n(x)$ and $N_n(x)$ are alternatingly increasing.

If $f(x)$ is symmetric and $\deg f(x)=n$, then it can be expanded as
$$f(x)=\sum_{k=0}^{\lrf{{n}/{2}}}\gamma_kx^k(1+x)^{n-2k},$$ and it is said to be {\it $\gamma$-positive}
if $\gamma_k\geq 0$ for $0\leq k\leq \lrf{\frac{n}{2}}$.
If $f(x)$ is $\gamma$-positive, then $f(x)$ is unomidal and symmetric.
Following~\cite[Definition~15]{Ma2001},
if $$f(x)=(1+x)^\nu\sum_{k=0}^{n}\lambda_kx^k(1+x^2)^{n-k}$$ and $\lambda_k\geq 0$ for all $0\leq k\leq n$,
then we say that $f(x)$ is {\it semi-$\gamma$-positive}, where $\nu=0$ or $\nu=1$.
Therefore, if $f(x)$ is semi-$\gamma$-positive, then
\begin{align*}
f(x)&=(1+x)^\nu\left(\sum_{k=0}^{\lrf{n/2}}\lambda_{2k}x^{2k}(1+x^2)^{n-2k}+\sum_{k=0}^{\lrf{(n-1)/2}}\lambda_{2k+1}x^{2k+1}(1+x^2)^{n-1-2k}\right)\\
&=(1+x)^\nu(f_1(x^2)+xf_2(x^2)),
\end{align*}
and both $f_1(x)$ and $f_2(x)$ are $\gamma$-positive (see~\cite[Proposition~16]{Ma2001}).

We now recall a very recent result on the flag ascent-plateau polynomials.
\begin{proposition}[{\cite[Theorem~19]{Ma2001}}]\label{Lnx-prop}
The polynomial $L_n(x)$ is semi-$\gamma$-positive. More precisely, for $n\geq 1$,
we have $$L_n(x)=\sum_{k=1}^nL_{n,k}x^k(1+x^2)^{n-k},$$
where the numbers $L_{n,k}$ satisfy the recurrence relation
\begin{equation}\label{Lnk-recu}
L_{n+1,k}=kL_{n,k}+L_{n,k-1}+4(n-k+2)L_{n,k-2},
\end{equation}
with the initial conditions $L_{0,0}=1$ and $L_{n,0}=0$ for $n\geq 1$.
\end{proposition}

It follows from~\eqref{Lnx-express} that
\begin{equation}\label{MnxNnx}
M_n(x)=L_n^E(x)+L_n^O(x),~N_n(x)=L_n^E(x)+xL_n^O(x).
\end{equation}
Since $\deg L_n(x)=2n-1$, we have $\deg L_n^E(x)=\deg L_n^O(x)=n-1$.
Note that $L_n^E(0)=0$ and $L_n^O(0)=1$.
By using Proposition~\ref{Lnx-prop}, we get that both $L_n^E(x)$ and $L_n^O(x)$ are $\gamma$-positive for any $n\geq 1$.
More precisely, we have
\begin{equation}\label{LnELnO}
\left\{
  \begin{array}{l}
L_n^E(x)=\sum_{\sigma\in\mathcal{Q}_n^+}x^{{\rm ap\,}(\sigma)}=\sum_{k=1}^{\lrf{n/2}} L_{n,2k} x^k(1+x)^{n-2k},, \\
L_n^O(x)=\sum_{\sigma\in\mathcal{Q}_n^-}x^{{\rm ap\,}(\sigma)}=\sum_{k=0}^{\lrf{(n-1)/2}} L_{n,2k+1}x^k(1+x)^{n-1-2k}.
  \end{array}
\right.
\end{equation}
In conclusion, we now present the fifth main result of this paper.
\begin{theorem}
For any $n\geq 1$, both $M_n(x)$ and $N_n(x)$ are alternatingly increasing, and so they are unimodal with modes in the middle.
\end{theorem}
\begin{proof}
It follows from~\eqref{LnELnO} that both $L_n^E(x)$ and $L_n^O(x)$ are symmetric and unimodal. When $n=2m+1$,
assume that
\begin{align*}
L_{2m+1}^E(x)&=\ell_1x+\ell_2x^2+\cdots+\ell_{m-1}x^{m-1}+\ell_mx^m+\ell_{m}x^{m+1}+\ell_{m-1}x^{m+2}+\cdots \ell_2x^{2m-1}+\ell_1x^{2m},\\
L_{2m+1}^O(x)&=1+\widetilde{\ell}_1x+\widetilde{\ell}_2x^2+\cdots+\widetilde{\ell}_{m-1}x^{m-1}+\widetilde{\ell}_mx^m+\widetilde{\ell}_{m-1}x^{m+1}+\cdots+\widetilde{\ell}_1x^{2m-1}+x^{2m}.
\end{align*}
Then $M_{2m+1}(x)=L_{2m+1}^E(x)+L_{2m+1}^O(x)=\sum_{i=0}^{2m}M_{2m+1,i}x^i$,
where
$$M_{2m+1,i}=\left\{
  \begin{array}{ll}
    1, & \hbox{if $i=0$;} \\
    \ell_i+\widetilde{\ell}_i, & \hbox{if $1\leq i\leq m$;} \\
    \ell_{2m-i+1}+\widetilde{\ell}_{2m-i}, & \hbox{if $m+1\leq i\leq 2m-1$;} \\
    \ell_1+1, & \hbox{if $i=2m$.}
  \end{array}
\right.$$
It is clear that $1\leq \ell_1+1\leq \ell_1+\widetilde{\ell}_1\leq \ell_2+\widetilde{\ell}_1\leq \cdots\leq \ell_m+\widetilde{\ell}_m$, i.e.,
$$M_{2m+1,0}\leq M_{2m+1,2m}\leq M_{2m+1,1}\leq M_{2m+1,2m-1}\leq \cdots\leq M_{2m+1,m}.$$
Note that $N_{2m+1}(x)=L_{2m+1}^E(x)+xL_{2m+1}^O(x)=\sum_{i=0}^{2m+1}N_{2m+1,i}x^i$,
where
$$N_{2m+1,i}=\left\{
  \begin{array}{ll}
    0, & \hbox{if $i=0$;} \\
    \ell_1+1, & \hbox{if $i=1$;} \\
    \ell_i+\widetilde{\ell}_{i-1}, & \hbox{if $2\leq i\leq m$;} \\
    \ell_{2m-i+1}+\widetilde{\ell}_{2m-i+1}, & \hbox{if $m+1\leq i\leq 2m$;} \\
    1, & \hbox{if $i=2m+1$.}
  \end{array}
\right.$$
It is clear that $0<1\leq \ell_1+1\leq \ell_1+\widetilde{\ell}_1\leq \ell_2+\widetilde{\ell}_1\leq \cdots\leq \ell_m+\widetilde{\ell}_m$, i.e.,
$$N_{2m+1,0}\leq N_{2m+1,2m+1}\leq N_{2m+1,1}\leq N_{2m+1,2m}\leq \cdots\leq N_{2m+1,m+1}.$$
Therefore, both $M_{2m+1}(x)$ and $N_{2m+1}(x)$ are alternatingly increasing, the mode of $M_{2m+1}(x)$ is $m$ and that of $N_{2m+1}(x)$ is $m+1$.
In the same way, one can verify that
both $M_{2m}(x)$ and $N_{2m}(x)$ are alternatingly increasing with the mode $m$. This completes the proof.
\end{proof}
\subsection{The up-down run polynomials for symmetric group}
\hspace*{\parindent}

Let $\pi\in\mathcal{S}_n$. The {\it up-down runs} of a permutation $\pi\in\mathcal{S}_n$ are the alternating runs of $\pi$ endowed with a 0
in the front~(see~\cite{Taylor03,Ma2001}). Let ${\rm udrun\,}(\pi)$ denote the number of up-down runs of $\pi$.
The {\it interior peak} number and {\it left peak} number of $\pi$ are respectively defined by
\begin{align*}
{\rm ipk\,}(\pi)&=\#\{i\in[2,n-1]:~\pi(i-1)<\pi(i)>\pi(i+1)\},\\
{\rm lpk\,}(\pi)&=\#\{i\in[n-1]: \pi(i-1)<\pi(i)>\pi(i+1),~\pi(0)=0\}.
\end{align*}
Define
$$W_n(x)=\sum_{\pi\in\mathcal{S}_n}x^{{\rm ipk\,}(\pi)},~\overline{W}_n(x)=\sum_{\pi\in\mathcal{S}_n}x^{{\rm lpk\,}(\pi)}.$$
The polynomials $W_n(x)$ and $\overline{W}_n(x)$ satisfy the recurrence rleations
$$W_{n+1}(x)=(nx-x+2)W_n(x)+2x(1-x)W_n'(x),$$
$$\overline{W}_{n+1}(x)=(nx+1)\overline{W}_n(x)+2x(1-x)\overline{W}_n'(x),$$
with $W_1(x)=\overline{W}_1(x)=1$ (see~\cite{Ma121,Petersen06}).
Note that $\deg \overline{W}_n(x)\geq \deg W_n(x)$. Then we
set $$p_n(x)=nx+1,~q_n(x)=2x(1-x),~r_n(x)=0.$$ Note that
$p_n(x)+\frac{1}{2x}q_n(x)=nx-x+2$. Then by using Theorem~\ref{mathm01}, we can define
\begin{align*}
\alpha_n(x)&=p_n(x^2)+\frac{1}{x}r_n(x^2)=nx^2+1,~
\beta_n(x)=\frac{1}{2x}q_n(x^2)=x(1-x^2).
\end{align*}
Therefore, we immediately get the following result, which has been proved in~\cite{Ma121}.
\begin{proposition}
Let $\{R_n(x)\}_{n\geq 1}$ be a sequence of polynomials defined by
\begin{equation}\label{Rnx01}
R_{n+1}(x)=(nx^2+1)R_n(x)+x(1-x^2)R_n'(x),
\end{equation}
with $R_1(x)=1+x$. Then $R_n(x)=\overline{W}_n(x^2)+xW_n(x^2)$.
\end{proposition}

The {\it up-down run polynomials} $T_n(x)$ are defined by $$T_n(x)=\sum_{\pi\in\mathcal{S}_n}x^{{\rm udrun\,}(\pi)}.$$
The polynomials $T_n(x)$ satisfy the recurrence relation
\begin{equation}\label{Rnx02}
T_{n+1}(x)=x(nx+1)T_{n}(x)+x\left(1-x^2\right)T_{n}'(x),
\end{equation}
with initial conditions $T_0(x)=1$ and $T_1(x)=x$ (see~\cite{Ma132,Stanley08}).
Comparing~\eqref{Rnx01} with~\eqref{Rnx02}, it is easy to check that $$R_n(x)=\frac{1+x}{x}T_{n}(x).$$

Set $\mathcal{S}_n^+=\{\pi\in\mathcal{S}_n:~\pi(n-1)>\pi(n)\}$ and $\mathcal{S}_n^-=\{\pi\in\mathcal{S}_n:~\pi(n-1)<\pi(n)\}$.
We define
$$T_n^E(x)=\sum_{\pi\in\mathcal{S}_n^+}x^{{\rm lpk\,}(\pi)},~T_n^O(x)=\sum_{\pi\in\mathcal{S}_n^-}x^{{\rm lpk\,}(\pi)}.$$
By definition, we get the following result.
\begin{proposition}\label{PropUnx}
For $n\geq 1$, we have
\begin{equation*}
\left\{
  \begin{array}{l}
T_n(x)=T_n^E(x^2)+xT_n^O(x^2),\\
\overline{W}_n(x)=T_n^E(x)+T_n^O(x).
  \end{array}
\right.
\end{equation*}
\end{proposition}

From~\eqref{Rnx02}, we see that $\alpha_n(x)=nx^2+x,~\beta_n(x)=x(1-x^2)$ and $\beta_n^e(x)=0$.
Using~\eqref{dualrelation01}, we obtain $x(nx^2+x)=xp_n(x^2)+r_n(x^2)$ and $2x^2(1-x^2)=q_n(x^2)$.
Hence $$p_n(x)=nx,~q_n(x)=2x(1-x),~r_n(x)=x,$$
$$u_n(x)=nx+1-x=(n-1)x+1,~v_n(x)=2x(1-x),~w_n(x)=1.$$

By using~\eqref{Rnx02} and~\cite[Theorem~2]{Ma2008}, we see that $T_n(x)$ have only real nonpositive zeros.
Combining Theorem~\ref{mathm01} and Theorem~\ref{Hermite}, we obtain the sixth main result of this paper.
\begin{theorem}
For $n\geq1$, we have
\begin{equation}\label{Tnx-recu}
\left\{
  \begin{array}{l}
    T_{n+1}^E(x)=nxT_n^E(x)+2x(1-x)\frac{d}{dx}{T_n^E}(x)+xT_n^O(x), \\
    T_{n+1}^O(x)=((n-1)x+1)T_n^O(x)+2x(1-x)\frac{d}{dx}T_n^O(x)+T_n^E(x),
  \end{array}
\right.
\end{equation}
with the initial conditions $T_1^E(x)=0$ and $T_1^O(x)=1$.  Moreover, both $T_n^E(x)$ and $T_n^O(x)$ have only real negative zeros and $T_n^O(x)\prec T_n^E(x)$.
\end{theorem}
\subsection{Alternating run polynomials for signed permutations}
\hspace*{\parindent}

Let $\pi\in{\mathcal S}^B_n$. The peak number and valley number of $\pi$ are respectively defined by
\begin{align*}
{\rm pk\,}(\pi)&=\#\{i\in[n-1]:~\pi(i-1)<\pi(i)>\pi(i+1),~\pi(0)=0\},\\
{\rm val\,}(\pi)&=\#\{i\in[n-1]: \pi(i-1)>\pi(i)<\pi(i+1),~\pi(0)=0\}.
\end{align*}
Recall that $C_n^+=\{\pi\in{\mathcal S}^B_n:~\pi(1)>0\}$.
We define
\begin{equation}\label{PnxVnx-def}
\begin{split}
U_n(x)&=\sum_{\pi\in C_n^+}x^{{\rm pk\,}(\pi)},~
V_n(x)=\sum_{\pi\in C_n^+}x^{{\rm val\,}(\pi)}.
\end{split}
\end{equation}

According to~\cite[Corollary~7]{Chow14}, the polynomials $U_n(x)$ and $V_n(x)$ satisfy the following system of Eulerian recurrences:
\begin{equation*}
\left\{
  \begin{array}{l}
U_{n+1}(x)=(2nx+1)U_{n}(x)+4x(1-x)U_{n}'(x)+xV_{n}(x),\\
V_{n+1}(x)=(2nx-2x+3)V_{n}(x)+4x(1-x)V_{n}'(x)+U_{n}(x),
  \end{array}
\right.
\end{equation*}
with $U_{0}(x)=1$ and $V_{0}(x)=0$. Note that $\deg U_n(x)\geq \deg V_n(x)$. Thus $(U_n(x),V_n(x))$ is a Eulerian pair.
Put
$$p_n(x)=2nx+1,~q_n(x)=4x(1-x),~r_n(x)=x,$$
$$u_n(x)=2nx-2x+3,~v_n(x)=4x(1-x),~w_n(x)=1.$$
Then $\alpha_n(x)=p_n(x^2)+\frac{1}{x}r_n(x^2)=2nx^2+x+1$ and
$\beta_n(x)=\frac{1}{2x}q_n(x^2)=2x(1-x^2)$.
Then by using Theorem~\ref{mathm01}, we immediately get the following result.
\begin{proposition}
Let $\{H_n(x)\}_{n\geq 0}$ be a sequence of polynomials defined by
\begin{equation}\label{Rnx01}
H_{n+1}(x)=(2nx^2+x+1)H_n(x)+2x(1-x^2)H_n'(x),
\end{equation}
with $H_0(x)=1$. Then $H_n(x)=U_n(x^2)+xV_n(x^2)$.
\end{proposition}

Let ${\rm altrun\,}(\pi)={\rm pk\,}(\pi)+{\rm val\,}(\pi)$ be the number of alternating runs of $\pi$.
The alternating run polynomials for signed permutations are given as follows:
$$\widetilde{H}_n(x)=\sum_{\pi\in C_n^+}x^{{\rm altrun\,}(\pi)}.$$
Zhao~\cite{Zhao11} showed that
$\widetilde{H}_{n+1}(x)=(2nx^2+3x-1)\widetilde{H}_{n}(x)+2x\left(1-x^2\right)\widetilde{H}_{n}'(x)$ for $n\geqslant 1$,
with $\widetilde{H}_1(x)=x$. It is routine to verify that $$H_n(x)=\frac{1+x}{x}\widetilde{H}_n(x)$$ for $n\geq 1$, which has been proved in~\cite[Theorem~8]{Chow14}.
\section{Concluding remark}
In this paper, we consider the combinatorial aspects of the Hermite-Biehler decompositions of several enumerative polynomials.
Let $\{f_n(x)\}_{n\geq 0}$ be a sequence of polynomials with nonnegative coefficients. Suppose that
\begin{equation}\label{recufnx}
f_{n+1}(x)=\left(a_1n+a_2+(b_1n+b_2)x+(c_1n+c_2)x^2\right)f_n(x)+dx(1-x^2)f_n'(x).
\end{equation}
where $a_1,a_2,b_1,b_2,c_1,c_2,d\in\mathbb{R}$. Then
$$\alpha_n(x)=a_1n+a_2+(b_1n+b_2)x+(c_1n+c_2)x^2,~\beta_n(x)=dx(1-x^2).$$
It follows from~\eqref{dualrelation01} that
$$p_n(x)=a_1n+a_2+(c_1n+c_2)x,~q_n(x)=2dx(1-x),~r_n(x)=(b_1n+b_2)x.$$
By using Theorem~\ref{mathm01}, we obtain
$$u_n(x)=a_1n+a_2+d+(c_1n+c_2-d)x,~v_n(x)=2dx(1-x),~w_n(x)=b_1n+b_2,$$
and then we can derive the recurrence system of the polynomials $f^E(x)$ and $f^O(x)$.

Besides the polynomials discussed in Section~\ref{Section02}, many other enumerative polynomials also satisfy the recurrence~\eqref{recufnx}, see~\cite{Barbero14,Zhu,Zhuang17} for instance.
We end this paper by giving an example. Following~\cite[Definition~1]{Aval13}, a tree-like tableau is a Ferrers diagram where each cell
contains either 0 or 1 point with some constraints. The symmetric tableaux are tree-like tableaux which are invariant with respect to reflection
through the main diagonal of their diagram.
Let $b(n,k)$ be the number of symmetric tableaux of size $2n+1$ with $k$ diagonal cells, and let $b_n(x)=\sum_{k=1}^{n+1}b(n,k)x^k$.
It follows from~\cite[Proposition~18]{Aval13} that $$b_{n+1}(x)=(n+1)x(1+x)b_n(x)+x(1-x^2)b_n'(x),$$
with the initial condition $b_0(x)=x$. By using the recurrence system of the polynomials $b_n^E(x)$ and $b_n^O(x)$,
one can easily derive that $b_n^E(1)=b_n^O(1)=2^{n-1}n!$ for $n\geq 1$. We leave the details to the reader.
Thus it may be interesting to further explore properties of $b_n^E(x)$ and $b_n^O(x)$.
\section*{Acknowledgements.}
This work is supported by NSFC (12071063,11571235) and
NSC (108-2115-M-017-005-MY2,~107-2115-M-001-009-MY3).

redpajama_set_name: RedPajamaArXiv


Salman Rushdie and Padma Lakshmi managed a colourful "Monsoon wedding in Manhattan" on Saturday night, at which the bride wore a purple sari with a daring, backless choli and the groom looked fetching in a dark sherwani with a silver-grey dupatta.
The wedding and party were held on the 14th floor of a New York studio and was attended by 275 guests, among them many of Rushdie's British literary friends who had supported him through his lean fatwa years. Hollywood showbiz and Indian relatives, celebrities and friends came, too.
In some ways, the ceremony was remarkably Hindu. There had been rumours that actress and model Padma would break Indian tradition by wearing a white sari but the Chennai girl dazzled in south Indian purple.
"She looked absolutely fabulous — and very happy," said the almost-lost-for-words New York-based Indian photographer, Jay Mandal, who was given special access on behalf of The Telegraph by the happy couple.
Had Rushdie won the Nobel Prize, he would not look happier.
It was the fourth wedding for Rushdie and Padma's first. The age difference appeared not to matter one bit. The 56-year-old bridegroom and the bride, 25 years his junior, garlanded each other under a makeshift mandap constructed with rajanigandha flowers and took vows from the Rig Veda. The bridegroom also arrived hand in hand with a little child — his own son, Milan, from his third marriage to British writer Elizabeth West (with whom he wrote an anthology of Indian writing).
In recognition of Rushdie's literary connections, the ceremony also included a reading from Shakespeare.
There was an Islamic touch as well, though from the religion's more mystical end. The well-known Indian vocalist, Marina Alam, a disciple of Pandit Jasraj, gave a rendering of Sufi songs. Not only did Rushdie slide a wedding ring on to his beloved's finger but he also knelt down to put a shoe on Padma's feet — the Prince Charming metaphorically finding his Cinderella.
Guests started arriving just before 5 pm at the Peter White Studio at 601 West 26th Street. After the marriage vows were exchanged, Padma slipped down to a huge trailer parked outside and took nearly an hour to change into a white, embroidered Luca Luca outfit with a low-cut top. Rushdie also swapped his dark sherwani for a cream one. There was a touch of Bollywood razzmatazz about the whole affair, which is appropriate because the Indian scene is currently very much in vogue on the back of Bom- bay Dreams in Broadway.
Although security was tight throughout, Mandal said he congratulated Rushdie on his marriage but found the 56-year-old author of Midn- ight's Children "lost in a haze of happiness".
Among Indian guests, Mandal spotted film director Ismail Merchant; Sabeer Bhatia, the inventor of Hotmail (who has himself been scouting around for a suitable bride from Bollywood for some years), Anupam Kher's actor son, Sikandar, and the Mumbai-based film director, Apoorva Lakhia, who worked as an assistant on Boom.
Although Boom, starring Padma, flopped, it is Lakhia's intention to direct her in a movie version of a 1997 short story by Rushdie, The Firebird's Nest, for which the author himself will write the script.
Rushdie's British friends included journalist couple Harold Evans and Tina Brown — Rushdie met Padma in 1999 at a party hosted by Brown at the Statue of Liberty for the now defunct Talk magazine.
Among other Brits were Somali-born model Iman; William Dalrymple, author of White Mughals — he and Rushdie did a joint literary session at Christie's in London last year — and his wife, Olivia; the human rights lawyer, Geoffrey Robertson, QC; columnist Christopher Hitchens and his wife, Carol Blue; and the humourist, Kathy Lette, at whose apartment Rushdie recently held an all-woman Stag Night.
Among Americans, there were actors Steve Martin and Robert Klein, New York A-list socialite Anne Bass and rocker Lou Reed. The Greek royal family was also represented.
There was dancing till late, with food supplied by an Indian restaurant, Bukhara Grill.


Hi guys. 2014 has been a year of experimenting for me. Yeah experimenting. When I cut my hair last year, I felt so liberated and was no longer driven to grow my hair long. My main hair goal became healthy looking hair. This year, I have delved my hands in different techniques and products. Over my next few posts, I will be discussing the techniques and products I tried out in 2014, the ones I liked and the ones I didn't.
First up is moisturizers. Why? If you are like me, this has been the most difficult product for me to settle on my entire hair journey. I have tried different types from S-curl to ORS carrot oil to Elasta QP mango butter to Darcy Botanicals transitioning cream, etc. I decided to do a battle of moisturizers and I mentioned them in this post HERE.
Anyways, I have 'settled' on Hairveda Red Tea Satin Moisturizer as my best daily moisturizer for my relaxed hair because when used in LITTLE amounts on my roller set hair, it doesn't revert. My major qualm with this moisturizer is that when you apply it first, your hair doesn't feel soft. Give it like 12 hours and your hair feels like silk. Very strange phenomenon! I still like the Pura Body Sapote lotion but it makes my hair lose its silkiness. It is however nice on my daughter's natural hair.
Anyways, Happy New Year to everyone. Much love. Look out for MORE posts from me in the new year!! HUGS!


Studies on synaptogenesis in regenerating nerve and muscle preparations have illustrated that the cue or cues directing synapse formation are to be found in the ECM, or "basal lamina ghosts"1,2. Many putative signaling molecules have been tested, but agrin quickly emerged as the strongest candidate. Agrin was identified and named based upon its ability to aggregate AChRs into clusters on cultured myotubes, forming sites of postsynaptic differentiation that closely resembled the sites seen on developing muscle fibers in vivo3. Although this AChR clustering activity was detectable in protein homogenates from many sources, it was ultimately purified by a combination of biochemical and antibody approaches using ECM prepared from the T. californica electric organ. The cDNA sequence was determined shortly thereafter4,5. In 1990, Dr. U. J. McMahan put forward the "Agrin hypothesis," stating that agrin is the nerve-derived organizer of postsynaptic differentiation at the NMJ6.
The agrin hypothesis has been beset by various challenges in the last 15 years, but has largely proven correct. Initial alarm was caused by the finding that agrin was made by muscle as well as motor neurons, thus making it unclear how it could be the nerve-derived organizer of the NMJ. This quandary was quickly resolved with the discovery of alternatively spliced forms of agrin (Figure 1.2).
Figure 1.2. The Agrin Protein. Agrin is a heparan sulfate proteoglycan of almost 2,000 amino acids. At its N-terminus, agrin has nine follistatin-like repeats that also have homology with Kazal-type protease inhibitor domains. Also in the N-terminal half are the sites of glycosaminoglycan (GAG) addition. This post-translational modification is usually heparan sulfate, but may sometimes include chondroitin sulfate. The SEA domain is proposed to mediate interactions with other extracellular O-linked glycoproteins. In the C-terminal half of the protein are four EGF-like repeats and three laminin-type globular G-domains (G). Agrin is alternatively spliced in its C-terminal half. The X-splice site is an alternative splice acceptor site in exon 20 of the agrin gene of unknown significance. The Y-splice site includes an exon of just 12 base pairs, encoding four amino acids. Inclusion of these amino acids confers heparin-binding activity to agrin. The Z-splice site involves two exons of 24 and 33 bp, resulting in the inclusion of eight, 11, or the combined 19 amino acids. The inclusion of these amino acids is specific for neuronally expressed isoforms of this protein, and these amino acids are necessary for MuSK activation and the AChR clustering activity of agrin. The N-terminus of agrin has two variants resulting from alternative transcriptional start sites and thus different translational start sites. The LN form of agrin is encoded by two distinct exons before joining the common sequence, resulting in an isoform of the protein that has a signal peptide for secretion from the cell and a domain that interacts with the y1 chain of laminins for assembly into the ECM. This is the predominant nonneuronal form of agrin, and it is also expressed by motor neurons, where it becomes anchored into the ECM of the NMJ. The other isoform, SN agrin, is encoded by a single unique exon before its transcript rejoins the common sequence at the same point as LN agrin. SN agrin is the primary form of agrin found in the brain. The protein is a type-2 transmembrane protein that remains associated with the cell surface. Its role in the brain and on the surface of neurons is under investigation.
Isoforms containing an insertion at a C-terminal alternative splice site (called the Z+ splice forms) were found to be active at inducing AChR clusters in cultured muscle fibers, while isoforms that did not include these amino acids (Z- isoforms) were inactive7. Furthermore, the active isoforms were made only in the nervous system, while muscle and other non-neuronal tissues made the inactive isoforms8,9. Thus, while agrin is made by both nerves and muscles, only nerves made isoforms of the protein that were active in AChR clustering, refining but not disproving the agrin hypothesis.