text:
Triple Review: Alabama Series by Susan Sands
Posted on July 7, 2017 by rolopolobookblog
Cammie Laroux is back in Alabama—again.
Dragged back to her small town to help her mother recover from surgery while rescuing the family event planning business should be a cinch. Even for a disgraced television chef, right? Wrong.
Among the many secrets Cammie's family's been hiding is the fact that their historic home is falling down. Oh, and the man hired to restore the house, Grey Harrison, is the same high school and college love of her life who thrashed her heart and dreams ten years ago. Yeah, that guy.
Grey, a widower with a young daughter, has never stopped loving Cammie, and when they are face to face once again, the chemistry is off the charts. Cammie may be in full-blown denial, but letting go is no longer in Grey's vocabulary, even when winning Cammie's forgiveness and renovating their love may seem like an impossible build even for a master architect and carpenter.
As Cammie finds herself forgetting all the reasons she can't trust Grey or love again, he finds himself remembering all the reasons he wants her to stay with him in Alabama… forever.
Source: Purchase My Rating: 4½/5 star
Cammie Laroux never had any intention of returning to her home state of Alabama for an extended period. When she left, she did so under the worst of circumstances so coming home under similar circumstances isn't at all what she had hoped for. Her culinary career is in shambles, her former co-worker is ruining her reputation in a very loud and public way, her mother is having a dangerous yet necessary back surgery, and the family business needs a competent hand at the helm. Since she's currently out of a job, Cammie is back home in Alabama, again.
Cammie's hometown isn't exactly a metropolis so her hope of staying under the radar is dashed almost immediately. Out of bed after a long flight, too little sleep, no shower, and only in her underwear, Cammie throws open the curtains only to find her rat-bastard ex-boyfriend standing on a ladder and adoring the view. Not the way to stay off the radar. As if the day couldn't get any worse, as Cammie begins her assessment of the house/business, she discovers everything is much more of a mess than she was led to believe. The house, an historic site they rent out for all types of events and occasions is coming apart at the seams and apparently, her mother hired her ex to oversee the considerable and extensive repairs. Even more traumatizing is the kitchen and pantry, Cammie's happy place. It's virtually bare and the house has events coming up sooner rather than later. Time to go to town . . . . .
Town is just as Cammie expected, a hot bed of gossip and stares and Cammie is the main topic of conversation. All anyone can talk about is how Cammie's career literally went up in flames and the fact her ex is back in town and working on her home. What everyone, including Cammie wants to know is how she feels about her ex, especially considering the fact he is once again single. Trouble is, Cammie isn't single and she's having a hard time remembering that. Her ex was always hard to resist and now that she's learning more about what really happened all those years ago and what he's endured in the meantime, he's becoming even more irresistible. With her family encouraging her to forgive and forget and knowing she'll be in town for an extended period while her mother recuperates Cammie considers the possibility of trying to mend a bridge or two.
As the days turn into weeks, Cammie finds it harder and harder to resist her ex's charms and as the house starts coming together and business is picking up, she begins to think more and more about where her future truly lies. Of course, those kinds of decisions aren't easy to make and the universe has a way of throwing up one obstacle after another. In the way of obstacles, Cammie and her ex(?) face all kinds of crazy including Cammie's fiancé, her former colleague who is still causing problems, one emotionally fragile little girl, and some family secrets not a single person in the world saw coming.
The Bottom Line: After reading Again, Alabama I feel a little like I need to find a small southern town to live in as there seems to be a lot of crazy that goes down there. Cammie's story is just chock full of crazy and it's enough to make a sane person go, well, crazy! From one day to the next, Cammie doesn't know what she's going to be hit with which makes having her family and ex-turned-friend near such a comfort. It's that comfort which reals Cammie in and reminds her of what's important in life. There is a bit of everything in this read that greatly appeals to me including a feisty heroine, a sexy, sweet and slightly damaged man, the past coming back to haunt A LOT of people, and enough southern drama and charm to satisfy everyone!
Goodreads | Amazon | B&N | Kobo
Emma Laroux's a fallen Southern beauty queen whose past is barely whispered about in her small town. But the secrets and lies surrounding her scandal still haunt her, and something about Matthew Pope may hold the answers…if only she could put her finger on it.
Matthew Pope wonders what awful karmic thing he's done to land him in Podunk, Alabama. But when he sees Emma Laroux again after all this time, he knows he's still the only one who holds the key to unlocking the truth of her past…
Will a shared moment in time ten years ago threaten the best thing that's ever happened to them – each other?
Source: Netgalley and purchase My Rating: 4/5 star
By nearly every standard, Emma Laroux is stunning. She has poise, grace, charm, beauty, all the things one would expect a former Miss Alabama to have. One would also expect a woman such as Emma Laroux to have a handsome man on her arm, but that is something Emma Laroux most definitely does not have. In fact, it's been roughly a decade since Emma last had a serious boyfriend.
In college, the world was seemingly at Emma Laroux's feet until she quite unexpectedly gave up her Miss Alabama crown and walked away from her long-term boyfriend. With little to no explanation, Emma left behind everything she had worked so hard for leaving everyone confused by her actions. Though Emma knows she made the right decision at the time, all these years later, she still isn't completely clear on what led her to such a huge, life-altering decision. Things about that time are hazy and lost and time has helped her remember.
In the ten years since her abdication (is that what you call it when a beauty queen gives up her crown??) Emma has led a successful life and mostly fulfilled life. She's tremendously close to her family, she has a successful pageant-prep business, and big dog who loves her unconditionally. Every single (and some married!) man in town wants her, but they also know she isn't going to say yes to a date. That is, until Matthew Pope comes to town.
Matthew Pope is a man of the strictest control. After a near-disastrous childhood, he knows first-hand what a loss of control looks like. He watches his food intake, he controls his exercise routine, and he runs a tight ship at work. As a result, Matthew has risen far in his field and carved a nice life. Emma Laroux makes him want to give up his strict hold on control and, in the beginning choke the life out of her and her big dog. Trouble is, that frustration and aggravation are mixed with some serious desire and Matthew Pope is determined to have Emma Laroux. Since he's new to town, he doesn't know about Emma's no dating rule so he dives right on in the deep end and hopes for the best! What he gets is a whole lot of sexy and more trouble than he could have ever imagined.
While Emma has worked hard to ignore/forget her past, being in a new relationship means sharing and that's something Emma isn't at all capable of doing. Matthew makes her think about the future, but until she can fully understand her past, a future just doesn't seem possible. Adding to the frustration is her now married ex-boyfriend turned town mayor who seems awfully interested in Emma's current situation. In fact, his interest in she and Matthew borders on stalking and no one, including her ex seems to be able to explain the behavior. As the days pass by and Emma and Matthew become more serious, it becomes clear Emma's ex is tied to her hazy past, the abandonment of her crown, and even some of the strange events of the last ten years. Figuring everything out means exposing a whole host of secrets no one, including Matthew want exposed.
The Bottom Line: I tell you what, Ministry, Alabama is just a hot bed of drama and awful secrets. Isn't it awesome??? Emma's story is dramatic, sad, scary, and ultimately very sweet. She's lived alone for so long she's gotten used to the feeling and doesn't even realize it until Matthew walks into her life. Matthew reads as somewhat nerdy and wimpy in the beginning, but as the tale unfolds he proves to be nerdy, strong, and sexy. In short, exactly what Emma needs! The road for Emma and Matthew is strange, twisted, and dark which makes the ultimate HEA so very, very good. Because of Emma's close relationship with her family, they play a huge part in this read and they shine in she and Matthew's defense. The only element of this read which kept me from a higher star rating is Matthew's family which added absolutely nothing to this read. At the end of it all, I enjoyed Love, Alabama and look so very forward to returning to Ministry, Alabama to see what new, awful secrets are lurking and just waiting to be uncovered.
Everybody loves local attorney and favorite son, Ben Laroux. Well, at least everybody of the female persuasion–until he meets Sabine O'Connor. She loathes him and makes no secret of her feelings, even when he pours on his famous charm hoping to thank her for helping his family. Ben has never been told no, and if there's one thing he's never walked away from, it's a challenge.
Sabine hopes she's finally found peace and safety in the small town of Ministry. She's changed her name to escape her painful past and her shameful family secrets. Ben Laroux is a gorgeous and sexy complication she can't afford, but also can't resist. However, when her past threatens to derail her present and future, Ben might be the only man she can trust.
Is it possible to be a colossal ass and not know you're a colossal ass?? In Ben Laroux's case, the answer is a resounding yes! Trouble is, he is Ministry, Alabama's most beloved colossal ass which has allowed him to go through life completely ignorant of his behavior. It's hard to hide things from the town shrink though 😊
Sabine spends her days trying to help those who are at the end of their rope. She listens, counsels, and keeps the secrets of countless Ministry, Alabama residents. Which is precisely why she knows the kind of man Ben Laroux is without ever having to talk to him. With his ridiculous two date rules, Ben has left many women in Ministry wanting more than he is ever going to be capable of giving. As Sabine has figured out, Ben thinks he's doing everyone a solid by being up front and honest. He never leads anyone on and simply walks away when the dating limit has been hit. Yeah, Sabine doesn't need any part of that arrogance in her life. She's full up on that kind of crap!
Sabine and her mother are only in Ministry until Sabine can secure a divorce from her powerful, egomaniacal husband. For two years, they've hidden in Ministry hoping he will eventually come to his senses and grant Sabine the divorce she desperately wants. After all the lies, the cheating, and the relentless pursuit of power, Sabine wants out of the public spotlight and as far away from her husband as is humanly possible. In fact, Sabine is ready to move forward with her life and that can't happen as long as she's still tied to her past.
When she refuses to give him even an ounce of consideration, Ben decides to push and poke at Sabine until she agrees to give him a chance. What Sabine finds is a man who is clueless about his behavior, who's stunningly sexy, and under all the bravado, a genuinely good man who cares for her. Which is just about the time Sabine sabotages it all by telling Ben about her past and her struggle to break free. Though he is justifiably angered, Ben does what all Laroux's do, he turns to his family for advice. As always, the Laroux's come to the rescue (HOWARD!!!!) and Ben crafts a plan that will not only secure Sabine's divorce, but land her despicable husband right where he belongs, in prison! To be sure, the plan is risky and before all is said and done, it becomes downright dangerous!
The Bottom Line: I am still 100% a smitten kitten with Susan Sands' Alabama series!! Of all the Ministry couples, Ben and Sabine are by far the most formidable. With her keen intelligence and insight into human nature and Ben's intelligence, determination, and kind nature, there's not much of anything or anyone who can stand in their way. Well, they can try 😊 Once again, Sands involves the entire Laroux family (and much of Ministry) which makes the read so much better, far more entertaining. Though I can't say for sure this is the end of the Alabama series, I can say, if it is, Ben and Sabine's story would be a fitting ending as it couples up all the Laroux kids and completes the family. With that being said, there are still a few characters hanging about (Sabine's sister for example) who are just begging to have their story told. I'm not being subtle here, Susan Sands, I want more of the Alabama series. ASAP!!!
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2 Comments on "Triple Review: Alabama Series by Susan Sands"
Susan Sands
LOL! I don't usually comment on reviews, but I felt compelled here!! Thanks for taking the time to review ALL THREE books! That takes time and commitment, friend! You are correct about Rachel's need for her own story. Christmas, Alabama will be released October 5th (tentatively)!! "Smitten as a kitten." Love it!! Thanks again!!
Oh, WOW!!!! Thank you so much for breaking with tradition and commenting on my reviews. I really do mean it, I so enjoyed this series and am very much looking forward to more 🙂


In this video Daarken will show you how to use the custom rain brush that was created in the last tutorial to add some rain to an existing painting.
In this tutorial Daarken goes over one way to create a custom rain brush. Many times you will want to avoid using too many custom brushes because they tend to be very obvious if not used correctly, so make sure not to overdo anything when using custom brushes.


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Q: Android Studio stack trace referencing imaginary directory I am trying to debug a signed apk of my app but the stack trace references a directory that doesn't exist.
2021-07-06 20:54:54.460 11377-11377/? E/AndroidRuntime: FATAL EXCEPTION: main
Process: com.germsoft.bizqrd, PID: 11377
java.lang.ExceptionInInitializerError
    at c.b.a.b.b.a(Contact.java:137)

As you can see from the photo, the directory c.b.a.b.b.a doesn't exist so I don't know where to start debugging this. Am I missing something?

redpajama_set_name: RedPajamaStackExchange


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\section{Introduction}

Let $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_{\ell})$ be a partition of a positive integer $n$. The {\it Young diagram} of $\lambda$ is a collection of $n$ boxes in $\ell$ rows with $\lambda_i$ boxes in row $i$. For example, the Young diagram for $\lambda=(5,4,2)$ is below.

\begin{center}
\tiny{
\begin{ytableau}
~&~&~&~&~ \\
~&~&~&~ \\
~&~
\end{ytableau}}
\end{center}

Let the leftmost column be column 1. The box in row $i$ and column $j$ is said to be in position $(i,j)$. For the Young diagram of $\lambda$, the partition $\lambda '=(\lambda_1 ',\lambda_2 ',\dots, \lambda_{\lambda_1} ')$ is called the {\it conjugate} of $\lambda$, where $\lambda'_j$ denotes the number of boxes in column $j$. A partition whose conjugate is equal to itself is called {\it self-conjugate}.
For each box in its Young diagram, we define its {\it hook length} by counting the number of boxes directly to its right or below, including the box itself. Equivalently, for the box in position $(i,j)$, the hook length of $\lambda$ is defined by
$$h(i,j)=\lambda_i+\lambda'_j-i-j+1.$$
For a positive integer $t$, a partition $\lambda$ is called a {\it $t$-core} if none of its hook lengths are multiples of $t$. We use the notation of a $(t_1,...,t_p)$-core if it is simultaneously a $t_1$-core,\dots, and a $t_p$-core. See for details \cite{AL,Anderson,AHJ,GKS,JK,Johnson,Wang}.\\

For a set $S$ of positive integers, we say that $a$ is generated by $S$ if $a$ can be written as a non-negative linear combination of the elements of $S$. Let $P=P_S$ be the set of elements which are not generated by $S$, and let $(P,<_P)$ be a poset by defining the cover relation so that $a$ covers $b$ if and only if $a-b\in S$. For example, see Figure \ref{fig:p8} for the poset $P_{\{8,9,10\}}$. For the detailed explanation of poset, we refer the reader to \cite{AL,S2,YZZ}.

\begin{figure}[ht!]
\centering
\begin{tikzpicture}[scale=1]

\foreach \i in {1,2,3,4,5,6,7}
{\path (\i,0) coordinate (\i);
\node[above] at (\i) {\i};}

\foreach \i in {11,12,13,14,15}
{\path (-9+\i,1) coordinate (\i);
\node[above] at (\i) {\i};
\draw (\i) -- (-10+\i,0.45);
\draw (\i) -- (-9+\i,0.45);
\draw (\i) -- (-8+\i,0.45);
}

\foreach \i in {21,22,23}
{\path (-18+\i,2) coordinate (\i);
\node[above] at (\i) {\i};
\draw (\i) -- (-19+\i,1.45);
\draw (\i) -- (-18+\i,1.45);
\draw (\i) -- (-17+\i,1.45);
}

\path (4,3) coordinate (31);
\node[above] at (31) {31};
\draw (31) -- (3,2.45);
\draw (31) -- (4,2.45);
\draw (31) -- (5,2.45);
\end{tikzpicture}
\caption{The Hasse diagram of $P_{\{8,9,10\}}$}\label{fig:p8}
\end{figure}

For a poset $(P,<_P)$, a set $I\subset P$ is called a \emph{lower ideal} of $P$ if $a<_P b$ and $b\in I$ implies $a\in I$.
In \cite{Anderson}, Anderson gave a natural bijection between $t$-cores and lower ideals of a poset $P_{\{t\}}$. Moreover, she proved that for relatively prime positive integers $s$ and $t$, the number of $(s,t)$-cores has a nice closed formula by finding a bijection between $(s,t)$-cores and lattice paths from $(0,0)$ to $(s,t)$ consisting of north and east steps which stay above the diagonal.

\begin{thm}\cite{Anderson}\label{thm:anderson}
For relatively prime positive integers $s$ and $t$, the number of $(s,t)$-cores is
$$\frac{1}{s+t}\binom{s+t}{s}.$$
\end{thm}

Since the work of Anderson, the topic counting simultaneous cores has received growing attention. In \cite{FMS}, Ford, Mai, and Sze proved the following analog of Anderson's work.

\begin{thm}\cite{FMS}\label{thm:FMS}
For relatively prime positive integers $s$ and $t$, the number of self-conjugate $(s,t)$-cores is
$$\binom{\lfloor \frac{s}{2}\rfloor+\lfloor \frac{t}{2}\rfloor}{\lfloor \frac{s}{2}\rfloor}.$$
\end{thm}

An \emph{$(s,k)$-generalized Dyck path} is a path from $(0,0)$ to $(s,s)$ which stays above the diagonal and consists of the steps $N_k=(0,k)$, $E_k=(k,0)$, and $D_i=(i,i)$ for $1\leq i \leq k-1$. For example, an $(s,1)$-generalized Dyck path is a (classical) \emph{Dyck path of order} $s$. We say that an $(s,k)$-generalized Dyck path is \emph{symmetric} if its reflection about the line $y=s-x$ is itself. It is often observed that counting the number of simultaneous cores can sometimes be described as counting the number of different paths.

\begin{rem} \label{rem:symmetric} Let $s$ be a positive integer.
\begin{enumerate}
\item[1.] The number of $(s,s+1)$-cores is the $s$th Catalan number $C_s=\frac{1}{s+1}\binom{2s}{s}$ which counts the number of Dyck paths of order $s$.
\item[2.] The number of self-conjugate $(s,s+1)$-cores is $\binom{s}{\lfloor s/2 \rfloor}$ which counts the number of \emph{symmetric Dyck paths} of order $s$.
\end{enumerate}
\end{rem}

In \cite{AL}, Amdeberhan and Leven expand Anderson's result to $(s,s+1,\dots,s+k)$-cores.

\begin{thm}\cite{AL}\label{thm:AL}
The followings are equinumerous:
\begin{enumerate}
\item[(a)] The number of $(s,s+1,\dots,s+k)$-cores.
\item[(b)] The number of $(s,k)$-generalized Dyck paths.
\item[(c)] The number of lower ideals in $P_{\{s,s+1,\dots,s+k\}}$.
\end{enumerate}
\end{thm}

We note that $(s,2)$-generalized Dyck paths are equivalent to Motzkin paths of length $s$. From Theorem \ref{thm:AL}, one can obtain the following corollary.

\begin{cor} \label{cor:Motzkin}
For a positive integer $s$, the number of $(s,s+1,s+2)$-cores is
$$M_s=\sum_{i=0}\frac{1}{i+1}\binom{s}{2i} \binom{2i}{i},$$
the $s$th Motzkin number which counts the number of Motzkin paths of length $s$.
\end{cor}

We note that Yang, Zhong, and Zhou \cite{YZZ} proved Corollary \ref{cor:Motzkin} independently.\\

It is natural to ask whether the number of self-conjugate $(s,s+1,s+2)$-cores and the number of symmetric Motzkin paths of length $s$ are equinumerous from Remark \ref{rem:symmetric} and Corollary \ref{cor:Motzkin}. In this paper, we prove that these two quantities are equal by showing that they satisfy the same recurrence relation which will be proved in Seciton \ref{sec:counting}. Furthermore, we give a closed formula for these numbers.

\section{Poset structure for self-conjugate $(s,s+1,s+2)$-cores}\label{sec:poset}

In this section, we construct a poset whose lower ideals with some restrictions are corresponding to self-conjugate $(s,s+1,s+2)$-cores, and then give a simple diagram to visualize that poset.\\

For a partition $\lambda$, let $MD(\lambda)$ denote the set of main diagonal hook lengths. Therefore, $MD(\lambda)$ is a set of distinct odds when $\lambda$ is self-conjugate. In \cite{FMS}, authors gave a useful result for determining self-conjugate $t$-cores.

\begin{prop}\cite{FMS}\label{prop:FMS}
Let $\lambda$ be a self-conjugate partition. Then $\lambda$ is a $t$-core partition if and only if both of the following hold:
\begin{enumerate}
\item[(a)] If $h\in MD(\lambda)$ with $h>2t$, then $h-2t\in MD(\lambda)$.
\item[(b)] If $h_1,h_2\in MD(\lambda)$, then $h_1+h_2\not\equiv 0 \pmod{2t}$.
\end{enumerate}
\end{prop}

For a positive integer $s$, we consider an induced subposet of $P=P_{\{2s,2s+1,\dots,2s+4\}}$,
$$\tilde{P}_{\{s,s+1,s+2\}}=\{h\in P~:~s\not<_P h,~s+1\not<_P h,~s+2\not<_P h, \text{~and~} h \text{~is odd} \}.$$
We note that the poset $\tilde{P}_{\{s,s+1,s+2\}}$ is the disjoint union of two posets, say $Q$ and $R$, where $Q$ is the maximal induced subposet of $P$ of which minimal elements are odd integers less than $s$, and $R$ is the maximal induced subposet of $P$ of which minimal elements are odd integers $x$ such that $s+2<x<2s$. See Figures \ref{fig:tilde8} and \ref{fig:tilde9} for example.

\begin{figure}[ht!]
\centering
\begin{tikzpicture}[scale=0.77]

\foreach \i in {1,3,5,7,11,13,15}
{\path (\i,0) coordinate (\i);
\node[above] at (\i) {\i};}

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\node[above, gray!60] at (\i) {\i};}

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\draw[gray!60] (\i) -- (-18+\i,0.55);
\draw[gray!60] (\i) -- (-19+\i,0.55);
\draw[gray!60] (\i) -- (-20+\i,0.55);
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\draw[thick] (\i) -- (-18+\i,0.55);
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\draw (\i) -- (-20+\i,0.55);
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\foreach \i in {41,42,43,44,45,46,47}
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\node[above, gray!60] at (\i) {\i};
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\draw[gray!60] (\i) -- (-35+\i,2.05);
\draw[gray!60] (\i) -- (-36+\i,2.05);
\draw[gray!60] (\i) -- (-37+\i,2.05);
\draw[gray!60] (\i) -- (-38+\i,2.05);
}

\foreach \i in {61,62,63}
{\path (-54+\i,4.5) coordinate (\i);
\node[above, gray!60] at (\i) {\i};
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\draw[gray!60] (\i) -- (-53+\i,3.55);
\draw[gray!60] (\i) -- (-54+\i,3.55);
\draw[gray!60] (\i) -- (-55+\i,3.55);
\draw[gray!60] (\i) -- (-56+\i,3.55);
}

\end{tikzpicture}
\caption{The Hasse diagram of the induced subposet $\tilde{P}_{\{8,9,10\}}$ of $P_{\{16,17,18,19,20\}}$}\label{fig:tilde8}
\end{figure}

\begin{figure}[ht!]
\centering
\begin{tikzpicture}[scale=0.77]

\foreach \i in {1,3,5,7,13,15,17}
{\path (\i,0) coordinate (\i);
\node[above] at (\i) {\i};}

\foreach \i in {2,4,6,8,9,10,11,12,14,16}
{\path (\i,0) coordinate (\i);
\node[above, gray!60] at (\i) {\i};}

\foreach \i in {24,26,27,28,29,30,31,32,33,34}
{\path (-20+\i,1.5) coordinate (\i);
\node[above, gray!60] at (\i) {\i};
\draw[gray!60] (\i) -- (-18+\i,0.55);
\draw[gray!60] (\i) -- (-19+\i,0.55);
\draw[gray!60] (\i) -- (-20+\i,0.55);
\draw[gray!60] (\i) -- (-21+\i,0.55);
\draw[gray!60] (\i) -- (-22+\i,0.55);
}

\foreach \i in {23,25,35}
{\path (-20+\i,1.5) coordinate (\i);
\node[above] at (\i) {\i};
\draw[thick] (\i) -- (-18+\i,0.55);
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\draw[thick] (\i) -- (-20+\i,0.55);
\draw[gray!60] (\i) -- (-21+\i,0.55);
\draw (\i) -- (-22+\i,0.55);
}

\foreach \i in {45,46,47,48,49,50,51,52,53}
{\path (-40+\i,3) coordinate (\i);
\node[above, gray!60] at (\i) {\i};
\draw[gray!60] (\i) -- (-38+\i,2.05);
\draw[gray!60] (\i) -- (-39+\i,2.05);
\draw[gray!60] (\i) -- (-40+\i,2.05);
\draw[gray!60] (\i) -- (-41+\i,2.05);
\draw[gray!60] (\i) -- (-42+\i,2.05);
}

\foreach \i in {67,68,69,70,71}
{\path (-60+\i,4.5) coordinate (\i);
\node[above, gray!60] at (\i) {\i};
\draw[gray!60] (\i) -- (-58+\i,3.55);
\draw[gray!60] (\i) -- (-59+\i,3.55);
\draw[gray!60] (\i) -- (-60+\i,3.55);
\draw[gray!60] (\i) -- (-61+\i,3.55);
\draw[gray!60] (\i) -- (-62+\i,3.55);
}

\end{tikzpicture}
\caption{The Hasse diagram of the induced subposet $\tilde{P}_{\{9,10,11\}}$ of $P_{\{18,19,20,21,22\}}$}\label{fig:tilde9}
\end{figure}

Now, we restate Proposition \ref{prop:FMS} by using the poset we constructed.

\begin{prop}\label{prop:lower}
Let $\lambda$ be a self-conjugate partition. Then $\lambda$ is an $(s,s+1,s+2)$-core partition if and only if the set $MD(\lambda)$ is a lower ideal of $\tilde{P}_{\{s,s+1,s+2\}}$ with no elements $h_1,h_2$ such that $h_1+h_2\in \{2s,2s+2,2s+4\}$.
\end{prop}

\begin{example}  For a self-conjugate $(8,9,10)$-core partition $\lambda=(6,3,3,1,1,1)$, the set $MD(\lambda)=\{11,3,1\}$ of main diagonal hook lengths is a lower ideal of $\tilde{P}_{\{8,9,10\}}$ with no elements $h_1,h_2$ such that $h_1+h_2\in\{16,18,20\}$.
\end{example}

For convenience, we add dotted edges connecting elements $h_1$, $h_2$ in the Hasse diagram of $\tilde{P}_{\{s,s+1,s+2\}}$ with $h_1+h_2\in \{2s,2s+2,2s+4\}$ so that at most one end point of each dotted edge can be selected for the lower ideal corresponding to an $(s,s+1,s+2)$-core. From now on, we use the \emph{modified diagram} for $\tilde{P}_{\{s,s+1,s+2\}}$ as in Figure \ref{fig:modified8}.

\begin{figure}[ht!]
\centering
\begin{tikzpicture}[scale=0.6]

\foreach \i in {1,3,5,7}
{\path (\i,0) coordinate (\i);
\node[above] at (\i) {\i};}

\foreach \i in {11,13,15}
{\path (17-\i,-1.5) coordinate (\i);
\node[above] at (\i) {\i};}

\foreach \i in {21,23}
{\path (-18+\i,1.5) coordinate (\i);
\node[above] at (\i) {\i};
\draw (\i) -- (-16+\i,0.75);
\draw (\i) -- (-18+\i,0.75);
\draw (\i) -- (-20+\i,0.75);
}

\path (4,-2.25) coordinate (31);
\node[below] at (31) {31};
\draw (31)-- (15)
       (31) -- (13)
       (31) -- (11);

\foreach \i in {1,3,5}
\draw[dotted] (\i)-- (2,-0.75);

\foreach \i in {3,5,7}
\draw[dotted] (\i)-- (4,-0.75);

\foreach \i in {5,7}
\draw[dotted] (\i)-- (6,-0.75);

\end{tikzpicture}
\qquad \qquad \quad\begin{tikzpicture}[scale=0.6]

\foreach \i in {1,3,5,7}
{\path (\i,0) coordinate (\i);
\node[above] at (\i) {\i};}

\foreach \i in {13,15,17}
{\path (19-\i,-1.5) coordinate (\i);
\node[above] at (\i) {\i};}

\foreach \i in {23,25}
{\path (-20+\i,1.5) coordinate (\i);
\node[above] at (\i) {\i};
\draw (\i) -- (-18+\i,0.75);
\draw (\i) -- (-20+\i,0.75);
\draw (\i) -- (-22+\i,0.75);
}

\path (4,-2.25) coordinate (35);
\node[below] at (35) {35};
\draw (35)-- (15)
       (35) -- (13)
       (35) -- (17);

\foreach \i in {1,3,5}
\draw[dotted] (\i)-- (2,-0.75);

\foreach \i in {3,5,7}
\draw[dotted] (\i)-- (4,-0.75);

\foreach \i in {5,7}
\draw[dotted] (\i)-- (6,-0.75);

\end{tikzpicture}
\caption{The modified diagrams of $\tilde{P}_{\{8,9,10\}}$ and $\tilde{P}_{\{9,10,11\}}$}\label{fig:modified8}
\end{figure}

We note that
$$Q\cong P_{\{\lfloor \frac{s}{2}\rfloor+1,\lfloor \frac{s}{2}\rfloor+2,\lfloor \frac{s}{2}\rfloor+3 \}}\quad \text{and} \quad  R\cong P_{\{\lfloor \frac{s}{2}\rfloor,\lfloor \frac{s}{2}\rfloor+1,\lfloor \frac{s}{2}\rfloor+2 \}},$$
and therefore, $\tilde{P}_{\{2s,2s+1,2s+2\}}$ is equivalent to $\tilde{P}_{\{2s+1,2s+2,2s+3\}}$. Moreover, it is not hard to notice that the modified diagrams of $\tilde{P}_{\{2s,2s+1,2s+2\}}$ and $\tilde{P}_{\{2s+1,2s+2,2s+2\}}$ are also equivalent. Thus, we have the following proposition.

\begin{prop}\label{prop:evenodd}
For a positive integer $s$, the number of self-conjugate $(2s,2s+1,2s+2)$-cores are equal to the number of self-conjugate $(2s+1,2s+2,2s+3)$-cores.
\end{prop}

\section{Counting self-conjugate simultaneous core partitions}\label{sec:counting}

In this section, we give a formula for the number of symmetric Motzkin paths, and then show that the number of self-conjugate $(2s,2s+1,2s+2)$-cores and the number of symmetric Motzkin paths of length $s$ satisfy the same recurrence relation.

\subsection{Counting symmetric Motzkin paths}

For a fixed $i$, there are $C_i \binom{n}{2i}$ Motzkin paths with exactly $i$ up steps since there are $C_i$ Dyck paths and there are $\binom{n}{2i}$ ways to insert $n-2i$ flat steps into a Dyck path with $i$ up steps. We say that a Motzkin path of length $n$ is \emph{symmetric} if its reflection about the line $x=\frac{n}{2}$ is itself. Let $S_n$ denote the number of symmetric Motzkin paths of length $n$. For exmaple, $S_0=1,S_1=1,S_2=2,S_3=2,S_4=5$.

\begin{figure}[ht!]
\centering
\begin{tikzpicture}[scale=0.5]
\draw (0,0) -- (1,1) -- (2,0) -- (3,1) -- (4,0);
\draw[-|] (5+0,0) -- (5+1,1);
\draw[-|] (5+2,2) -- (5+3,1);
\draw (5+1,1) -- (5+2,2)
      (5+3,1) -- (5+4,0);
\draw (10+0,0) -- (10+1,0) -- (10+2,1) -- (10+3,0) -- (10+4,0);
\draw[-|] (15+0,0) -- (15+1,1) -- (15+2,1);
\draw (15+2,1)-- (15+3,1) -- (15+4,0);
\draw[-|] (20+0,0)-- (20+1,0);
\draw[-|] (20+1,0)-- (20+2,0);
\draw[-|] (20+2,0)-- (20+3,0);
\draw (20+3,0)-- (20+4,0);
\end{tikzpicture}
\caption{Symmetric Motzkin paths of length $4$}
\end{figure}

We note that the $(n+1)$st step of any symmetric Motzkin path of length $2n+1$ must be a flat step, and therefore, there is a natural bijection between symmetric Motzkin paths of length $2n+1$ and that of length $2n$ so that $S_{2n+1}=S_{2n}$.

Now, we count the number of symmetric Motzkin paths.

\begin{prop}\label{prop:symmetric}
The number of symmetric Motzkin paths of length $n$ is
$$S_n=\sum_{i\geq0} \binom{\lfloor \frac{n}{2}\rfloor}{i}\binom{i}{\lfloor \frac{i}{2} \rfloor}.$$
\end{prop}

\begin{proof}
It is enough to enumerate symmetric Motzkin paths of length $2n$. Suppose we are given a symmetric Dyck path with $i$ up steps, so that it has $2n-2i$ flat steps. To obtain a symmetric Motzkin path of length $2n$ with $i$ up steps, it is enough to consider inserting $n-i$ flat steps into the first half of the given symmetric Dyck path. Since there are $\binom{i}{\lfloor i/2\rfloor}$ symmetric Dyck paths with $i$ up steps as in Remark \ref{rem:symmetric}, and there are $\binom{n}{i}$ ways to insert flat steps, the number of symmetric Motzkin paths of length $2n$ with $i$ up steps is $\binom{n}{i}\binom{i}{\lfloor i/2 \rfloor}$. Therefore, $S_{2n}=S_{2n+1}=\sum_{i\geq0}\binom{n}{i}\binom{i}{\lfloor i/2 \rfloor}$.
\end{proof}

Now, we consider a recurrence relation of $S_{2n}$ involving $M_{n}$. For a symmetric Motzkin path $P=P_1P_2\cdots P_{2n}$ of length $2n$, where $P_i$ denotes the $i$th step, let $k\leq n$ be the largest number such that $P$ meets $x$-axis at $(k,0)$.
We note that if $k=n$, then both of $P_1P_2\cdots P_{n}$ and $P_{n+1}P_{n+2}\cdots P_{2n}$ are Motzkin paths of length $n$ which are symmetric to each other.
On the other hand, if $k<n$, then $P_{k+1}=U$, $P_{2n-k}=D$, the subpath $P_{k+2}P_{k+3}\cdots P_{2n-k-1}$ is a symmetric Motzkin path of length $2n-2k-2$, and both of two subpaths $P_1P_2\cdots P_k$ and $P_{2n-k+1}P_{2n-k+2}\cdots P_{2n}$ are Motzkin paths of length $k$ which are symmetric to each other. Hence, we have a relation between $S_{2n}$ and $M_n$:
\begin{equation} \label{eqn:re}
S_{2n}=M_n+\sum_{k=0}^{n-1}S_{2n-2k-2}M_{k}.
\end{equation}

Equation (\ref{eqn:re}) and a closed formula for $S_{2n}$ can also be found in the OEIS as A005773 \cite{OEIS}.

\subsection{Counting self-conjugate $(2s,2s+1,2s+2)$-cores}

The following lemma plays an important role to obtain a recurrence relation for the number of self-conjugate $(2s,2s+1,2s+2)$-cores.

\begin{lem} \label{lem:with2s-1}
Let $s$ be a positive integer.
The number of self-conjugate $(2s,2s+1,2s+2)$-cores $\lambda$ with $2s-1\in MD(\lambda)$ is equal to the number of self-conjugate $(2s-2,2s-1,2s)$-cores.
\end{lem}

\begin{proof}
By Proposition \ref{prop:lower}, there is a bijection between $(2s,2s+1,2s+2)$-cores $\lambda$ with $2s-1\in MD(\lambda)$ and lower ideals $I$ of $\tilde{P}_{\{2s,2s+1,2s+2\}}$ containing $2s-1$ and no elements $h_1,h_2$ such that $h_1+h_2\in\{4s,4s+2,4s+4\}$. Thus, it is enough to consider lower ideals of the first diagram in Figure \ref{fig:modified2s}.

To prove the lemma, we construct a bijection $\phi$ between lower ideals $I$ of the first diagram and lower ideals $J$ of the second diagram in Figure \ref{fig:modified2s}, since the second diagram is equivalent to the modified diagram of $\tilde{P}_{\{2s-2,2s-1,2s\}}$.

\begin{figure}[ht!]
\centering
\begin{tikzpicture}[scale=0.65]

\foreach \i in {1,3,5,7,9,11}
{\path (\i,0) coordinate (\i);}
\node[above] at (1) {$1$};
\node[above] at (3) {$3$};
\node[above] at (5) {$\cdots$};
\node[above] at (7) {$\cdots$};
\node[above] at (9) {$2s-3$};
\node[above] at (11) {$\mathbf{2s-1}$};

\path (2,1.5) coordinate (2);
\node[above] at (2) {$4s+5$};
\draw (2) -- (1,0.7);
\draw (2) -- (3,0.7);
\draw (2) -- (5,0.7);

\path (4,1.5) coordinate (4);
\node[above] at (4) {$\cdots$};
\draw (4) -- (3,0.7);
\draw (4) -- (5,0.7);
\draw (4) -- (7,0.7);

\path (6,1.5) coordinate (6);
\node[above] at (6) {$\cdots$};
\draw (6) -- (5,0.7);
\draw (6) -- (7,0.7);
\draw (6) -- (9,0.7);

\path (8,1.5) coordinate (6);
\node[above] at (6) {$6s-1$};
\draw (6) -- (7,0.7);
\draw (6) -- (9,0.7);

\path (4,3) coordinate (8);
\node[above] at (8) {$\cdots$};
\draw (8) -- (2,2.2);
\draw (8) -- (4,2.2);
\draw (8) -- (6,2.2);

\path (6,3) coordinate (10);
\node[above] at (10) {$\cdots$};
\draw (10) -- (4,2.2);
\draw (10) -- (6,2.2);
\draw (10) -- (8,2.2);

\foreach \i in {13,15,17,19,21}
{\path (24-\i,-1.5) coordinate (\i);}

\node[above] at (21) {$4s-1$};
\node[above] at (19) {$\cdots$};
\node[above] at (17) {$2s+7$};
\node[above,gray!60] at (15) {$2s+5$};
\node[above,gray!60] at (13) {$2s+3$};

\path (5,-2.25) coordinate (31);
\node[below] at (31) {$\cdots$};
\draw (31)-- (21)
       (17) -- (31) -- (19);

\foreach \i in {1,3,5}
\draw[dotted] (\i)-- (3,-0.9);

\foreach \i in {3,5,7}
\draw[dotted] (\i)-- (5,-0.9);

\foreach \i in {5,7,9}
\draw[dotted] (\i)-- (7,-0.9);

\foreach \i in {11}
\draw[dotted] (\i)-- (9,-0.9);

\foreach \i in {11}
\draw[dotted] (\i)-- (11,-0.9);

\end{tikzpicture}
\quad
\begin{tikzpicture}[scale=0.65]

\node[above] at (-0.4,0) {$\cong$};
\foreach \i in {1,3,5,7,9}
{\path (\i,0) coordinate (\i);}
\node[above] at (1) {$1$};
\node[above] at (3) {$3$};
\node[above] at (5) {$\cdots$};
\node[above] at (7) {$\cdots$};
\node[above] at (9) {$2s-3$};

\path (2,1.5) coordinate (2);
\node[above] at (2) {$4s+5$};
\draw[dotted] (2) -- (1,0.7);
\draw[dotted] (2) -- (3,0.7);
\draw[dotted] (2) -- (5,0.7);

\path (4,1.5) coordinate (4);
\node[above] at (4) {$\cdots$};
\draw[dotted] (4) -- (3,0.7);
\draw[dotted] (4) -- (5,0.7);
\draw[dotted] (4) -- (7,0.7);

\path (6,1.5) coordinate (6);
\node[above] at (6) {$\cdots$};
\draw[dotted] (6) -- (5,0.7);
\draw[dotted] (6) -- (7,0.7);
\draw[dotted] (6) -- (9,0.7);

\path (8,1.5) coordinate (6);
\node[above] at (6) {$6s-1$};
\draw[dotted] (6) -- (7,0.7);
\draw[dotted] (6) -- (9,0.7);

\path (4,3) coordinate (8);
\node[above] at (8) {$\cdots$};
\draw (8) -- (2,2.2);
\draw (8) -- (4,2.2);
\draw (8) -- (6,2.2);

\path (6,3) coordinate (10);
\node[above] at (10) {$\cdots$};
\draw (10) -- (4,2.2);
\draw (10) -- (6,2.2);
\draw (10) -- (8,2.2);

\foreach \i in {17,19,21}
{\path (24-\i,-1.5) coordinate (\i);}

\node[above] at (21) {$4s-13$};
\node[above] at (19) {$\cdots$};
\node[above] at (17) {$2s+7$};

\path (5,-2.25) coordinate (31);
\node[below] at (31) {$\cdots$};
\draw (31)-- (21)
       (17) -- (31) -- (19);

\foreach \i in {1,3,5}
\draw (\i)-- (3,-0.9);

\foreach \i in {3,5,7}
\draw (\i)-- (5,-0.9);

\foreach \i in {5,7,9}
\draw (\i)-- (7,-0.9);
\end{tikzpicture}

\caption{The modified diagram of $\tilde{P}_{\{2s,2s+1,2s+2\}}$ for ideals having  $2s-1$}\label{fig:modified2s}
\end{figure}

If $I$ is a lower ideal of the first diagram, then $I$ satisfies the following:
\begin{itemize}
\item $I$ contains $2s-1$.
\item For $4s+5\leq h\leq 6s-1$, $h\in I$ implies $h-4s-4, h-4s-2, h-4s\in I$.
\item For $2s+7\leq h\leq 4s-1$, $h\in I$ implies $4s-h, 4s-h+2, 4s-h+4\not\in I$.
\end{itemize}

Now, we construct a corresponding set $\phi(I)$ from $I$ as follows.

\begin{itemize}
\item For each $h\in I$ with $4s+5\leq h\leq 6s-1$, delete $h-4s-4, h-4s-2, h-4s$ from $I$.
\item For each $h\in I$ with $2s+7\leq h\leq 4s-1$, add $4s-h, 4s-h+2, 4s-h+4$ to $I$.
\item Delete $2s-1$ from the set.
\end{itemize}

Then $\phi(I)$ is a lower ideal of the poset structure defined by the second diagram in Figure \ref{fig:modified2s} and it is easy to check that $\phi$ is a bijection.
\end{proof}

\begin{example} For self-conjugate $(8,9,10)$-cores $\lambda$ such that $7\in MD(\lambda)$, let $I_1, I_2,\dots,I_{13}$ be their corresponding lower ideals of $\tilde{P}_{\{8,9,10\}}$.  Now, we list $I_i$ and $J_i=\phi(I_i)$ for $i=1,2,\dots,13$, where $\phi$ is the bijection defined in the proof of Lemma \ref{lem:with2s-1}.\\

$\begin{array}{lr}
I_1=\{7\}\\
I_3=\{3,7\}\\
I_5=\{1,3,7\}\\
I_7=\{3,5,7\}\\
I_9=\{1,3,5,7,21\}\\
I_{11}=\{3,5,7,23\} \\
I_{13}=\{7,15\}
\end{array}
$
$\begin{array}{lr}
J_1=\emptyset\\
J_3=\{3\}\\
J_5=\{1,3\} \\
J_7=\{3,5\}\\
J_9=\{21\}\\
J_{11}=\{23\} \\
J_{13}=\{1,3,5,15\}
\end{array}
$
\quad\qquad
$\begin{array}{lr}
I_2=\{1,7\}\\
I_4=\{5,7\}\\
I_6=\{1,5,7\}\\
I_8=\{1,3,5,7\}\\
I_{10}=\{1,3,5,7,21,13\}\\
I_{12}=\{1,3,5,7,23\}
\end{array}
$
$\begin{array}{lr}
J_2=\{1\}\\
J_4=\{5\}\\
J_6=\{1,5\}\\
J_8=\{1,3,5\}\\
J_{10}=\{21,23\}\\
J_{12}=\{1,23\}
\end{array}
$\\

We note that for each $i$, $I_i$ is an ideal of the first diagram and $J_i$ is an ideal of the second diagram of the following figure.

\begin{figure}[ht!]
\centering
\begin{tikzpicture}[scale=0.65]

\foreach \i in {5,7,9,11}
{\path (\i,0) coordinate (\i);}
\node[above] at (5) {$1$};
\node[above] at (7) {$3$};
\node[above] at (9) {$5$};
\node[above] at (11) {$\mathbf{7}$};

\path (6,1.5) coordinate (6);
\node[above] at (6) {$21$};
\draw (6) -- (5,0.7);
\draw (6) -- (7,0.7);
\draw (6) -- (9,0.7);

\path (8,1.5) coordinate (6);
\node[above] at (6) {$23$};
\draw (6) -- (7,0.7);
\draw (6) -- (9,0.7);

\foreach \i in {13,15,17}
{\path (24-\i,-1.5) coordinate (\i);}

\node[above] at (17) {$15$};
\node[above,gray!60] at (15) {$13$};
\node[above,gray!60] at (13) {$11$};

\foreach \i in {5,7,9}
\draw[dotted] (\i)-- (7,-0.8);

\foreach \i in {11}
\draw[dotted] (\i)-- (9,-0.8);

\foreach \i in {11}
\draw[dotted] (\i)-- (11,-0.8);

\end{tikzpicture}
\quad
\begin{tikzpicture}[scale=0.65]

\node[above] at (3.6,0) {$\cong$};
\foreach \i in {5,7,9}
{\path (\i,0) coordinate (\i);}
\node[above] at (5) {$1$};
\node[above] at (7) {$3$};
\node[above] at (9) {$5$};

\path (6,1.5) coordinate (6);
\node[above] at (6) {$21$};
\draw[dotted] (6) -- (5,0.7);
\draw[dotted] (6) -- (7,0.7);
\draw[dotted] (6) -- (9,0.7);

\path (8,1.5) coordinate (6);
\node[above] at (6) {$23$};
\draw[dotted] (6) -- (7,0.7);
\draw[dotted] (6) -- (9,0.7);

\foreach \i in {17}
{\path (24-\i,-1.5) coordinate (\i);}
\node[above] at (17) {$15$};

\foreach \i in {5,7,9}
\draw (\i)-- (7,-0.8);
\end{tikzpicture}
\caption{The modified diagram of $\tilde{P}_{\{8,9,10\}}$ for ideals $I$ with $7\in I$}\label{fig:exam}
\end{figure}

\end{example}

The following proposition is a generalization of Lemma \ref{lem:with2s-1}.

\begin{prop} \label{prop:no2s-1}
Let $s$ and $k$ be positive integers such that $k\leq s$.
\begin{enumerate}
\item[(a)] The number of self-conjugate $(2s,2s+1,2s+2)$-cores $\lambda$ satisfies that
$$2k-1\in MD(\lambda) \quad \text{and} \quad 2k+1,2k+3,\dots, 2s-1\not\in MD(\lambda)$$
is the number of self-conjugate $(2k-2,2k-1,2k)$-cores multiplied by $M_{s-k}$.
\item[(b)] The number of self-conjugate $(2s,2s+1,2s+2)$-cores $\lambda$ with
$1,3,\dots, 2s-1\not\in MD(\lambda)$ is $M_{s}$.
\end{enumerate}
\end{prop}
\begin{proof}
We consider the modified diagram of $\tilde{P}_{\{2s,2s+1,2s+2\}}$ with restrictions in Figure \ref{fig:modified2s2}.

\begin{figure}[ht!]
\centering
\begin{tikzpicture}[scale=0.72]

\foreach \i in {1,3,5,7,9,11,13,15,17,19,21}
{\path (\i,0) coordinate (\i);}
\node[above] at (1) {\small$1$};
\node[above] at (3) {\small$3$};
\node[above] at (5) {\small$\cdots$};
\node[above] at (7) {\small$\cdots$};
\node[above] at (9) {\small$2k-3$};
\node[above] at (11) {\small$\mathbf{2k-1}$};
\node[above,gray!60] at (13) {\small$2k+1$};
\node[above,gray!60] at (15) {\small$2k+3$};
\node[above,gray!60] at (17) {\small$\cdots$};
\node[above,gray!60] at (19) {\small$2s-3$};
\node[above,gray!60] at (21) {\small$2s-1$};

\path (2,1.5) coordinate (2);
\node[above] at (2) {\small$4s+5$};
\draw (2) -- (1,0.7);
\draw (2) -- (3,0.7);
\draw (2) -- (5,0.7);

\path (4,1.5) coordinate (4);
\node[above] at (4) {\small$\cdots$};
\draw (4) -- (3,0.7);
\draw (4) -- (5,0.7);
\draw (4) -- (7,0.7);

\path (6,1.5) coordinate (6);
\node[above] at (6) {\small$\cdots$};
\draw (6) -- (5,0.7);
\draw (6) -- (7,0.7);
\draw (6) -- (9,0.7);

\path (8,1.5) coordinate (6);
\node[above] at (6) {\small$4s+2k-1$};
\draw (6) -- (7,0.7);
\draw (6) -- (9,0.7);

\path (4,3) coordinate (8);
\node[above] at (8) {\small$\cdots$};
\draw (8) -- (2,2.2);
\draw (8) -- (4,2.2);
\draw (8) -- (6,2.2);

\path (6,3) coordinate (10);
\node[above] at (10) {\small$\cdots$};
\draw (10) -- (4,2.2);
\draw (10) -- (6,2.2);
\draw (10) -- (8,2.2);

\foreach \i in {111,113,115,117,119,121,123,125,127}
{\path (130-\i,-1.5) coordinate (\i);}

\node[above] at (127) {\small$4s-1$};
\node[above] at (125) {\small$\cdots$};
\node[above] at (123) {\small$4s-2k+7$};
\node[above] at (115) {\small$4s-2k-1$};
\node[above] at (113) {\small$\cdots$};
\node[above] at (111) {\small$2s+3$};

\path (5,-2.25) coordinate (31);
\node[below] at (31) {\small$\cdots$};
\draw (31)-- (127)
       (123) -- (31) -- (125);

\path (17,-2.25) coordinate (33);
\node[below] at (33) {\small$\cdots$};
\draw (33)-- (111)
       (113) -- (33) -- (115);

\foreach \i in {1,3,5}
\draw[dotted] (\i)-- (3,-0.9);

\foreach \i in {3,5,7}
\draw[dotted] (\i)-- (5,-0.9);

\foreach \i in {5,7,9}
\draw[dotted] (\i)-- (7,-0.9);

\foreach \i in {9,11,13}
\draw[dotted] (11)-- (\i,-0.9);

\foreach \i in {13,15,17}
\draw[dotted] (\i)-- (15,-0.9);

\foreach \i in {15,17,19}
\draw[dotted] (\i)-- (17,-0.9);

\foreach \i in {17,19,21}
\draw[dotted] (\i)-- (19,-0.9);
\end{tikzpicture}

\caption{The modified diagram of $\tilde{P}_{\{2s,2s+1,2s+2\}}$ with restrictions}\label{fig:modified2s2}
\end{figure}

\begin{enumerate}
\item[(a)] On the modified diagram with restrictions, the left-hand side of $2k-1$ is equivalent to the modified diagram of $\tilde{P}_{\{2k-2,2k-1,2k\}}$ as we showed in Lemma \ref{lem:with2s-1}, and the right-hand side of $2k-1$ is equivalent to the Hasse diagram of the poset $P_{\{2s-2k-2,2s-2k-1,2s-2k\}}$. Hence, by Corollary \ref{cor:Motzkin}, the number of lower ideals $I$ satisfying $2k-1\in I$ and $2k+1,2k+3,\dots,2s-1\not \in I$ is $M_{s-k}$ times the number of self-conjugate $(2k-2,2k-1,2k)$-cores.
\item[(b)] If $1,3,\dots, 2s-1\not\in I$, then $I$ is a lower ideal of a subposet of $\tilde{P}_{\{2s,2s+1,2s+2\}}$ which is equivalent to $P_{\{s,s+1,s+2\}}$. Hence, by Corollary \ref{cor:Motzkin}, the number of lower ideals $I$ with $1,3,\dots,2s-1\not \in I$ is $M_{s}$.
\end{enumerate}
\end{proof}

Now, we are ready to prove our main theorem.

\begin{thm}
For a positive integer $s$, the number of self-conjugate $(s,s+1,s+2)$-cores is
$$\sum_{i\geq0}\binom{\lfloor \frac{s}{2}\rfloor}{i}\binom{i}{\lfloor \frac{i}{2} \rfloor},$$
which counts the number of symmetric Motzkin paths of length $s$.
\end{thm}

\begin{proof}
 Let $a_s$ denote the number of self-conjugate $(s,s+1,s+2)$-cores. From Proposition \ref{prop:evenodd}, we have $a_{2s+1}=a_{2s}$. Hence, it is enough to show that $a_{2s}=S_{2s}$.

For $1\leq k \leq s$, let $A_k$ be the set of self-conjugate $(2s,2s+1,2s+2)$-cores $\lambda$ that satisfies
$$2k-1\in MD(\lambda) \quad \text{and} \quad 2k+1,2k+3,\dots, 2s-1\not\in MD(\lambda),$$ and let $A_0$ be the set of self-conjugate $(2s,2s+1,2s+2)$-cores $\lambda$ with $2i-1\not \in MD(\lambda)$ for $1\leq i\leq s$. Then, $A_0\cup A_1\cup \cdots \cup A_s$ is the set of self-conjugate $(2s,2s+1,2s+1)$-cores and
$$a_{2s}=|A_0|+|A_1|+\cdots+|A_s|.$$
From Proposition \ref{prop:no2s-1}, we have $|A_0|=M_s$ and $|A_k|=a_{2k-2}M_{s-k}$ for $1\leq k \leq s$, and therefore,
$$a_{2s}=M_s+\sum_{k=1}^{s}a_{2k-2}M_{s-k}=M_s+\sum_{k=0}^{s-1}a_{2s-2k-2}M_{k}.$$
Since the relation between $a_{2s}$ and $M_{s}$ is equivalent to (\ref{eqn:re}) and
$a_0=S_0=1$, we have come to a conclusion that $a_{2s}=S_{2s}=\sum\binom{s}{i}\binom{i}{\lfloor i/2 \rfloor}$ by Proposition \ref{prop:symmetric}.
\end{proof}

Encouraged by this success, we offer the following generalized conjecture.

\begin{conj} For given positive integers $s$ and $k$, the number of self-conjugate $(s,s+1,\dots,s+k)$-cores is equal to the number of symmetric $(s,k)$-generalized Dyck paths.
\end{conj}

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


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text:
Born - 24 May 1931, Paris, France
Sturdy, dark-haired actor of sternly cherubic features, piercing gaze, chubby build and no apparent neck. Often moustachioed and bearded, and born Michael Lonsdale-Crouch of Anglo-French parentage, his early years were spent in both England and Morocco, before returning to France, where he began acting on radio, before making a film debut there in C'est Arrive a Aden (1956). His command of both English and French enabled him to appear in international films, billed either Michel or Michael, before he leapt to greater prominence hunting Edward Fox's assassin in The Day of the Jackal in 1973. Scores of film appearances, frequently in sinister roles, include those in La Main Chaude (1959), The Trial (1962), Behold a Pale Horse (1964), Hail! Mafia (1965), The Bride Wore Black (1967), Stolen Kisses (1968), Le Souffle au Coeur/Dearest Love (1971), The Phantom of Liberty (1974), The Romantic Englishwoman (1975), Mr Klein (1976), The Left-Handed Woman (1977), The Passage (1978), Moonraker (1979), Chariots of Fire (1981), Enigma (1982), The Holcroft Covenant (1985), The Name of the Rose (1986), Souvenir (1987), The Madonna Man (1988), Ma Vie est un Enfer (1991), The Remains of the Day (1993), Jefferson in Paris (1995), Ronin (1998), Les Acteurs (2000), The Mystery of the Yellow Room (2003), Bye Bye Blackbird (2004), Munich (2005), Goya's Ghosts (2006), Agora (2009), Of Gods and Men (2010), Le Renard Jaune (2013) and Les Premiers, les Derniers (2016), his last. He also wrote a number of novels.

redpajama_set_name: RedPajamaCommonCrawl


At Beechworth Secondary College, we have a system of care, involving form teachers, year level coordinators, a student wellbeing coordinator, a school nurse and classroom teachers.
At Year 7, our Peer Support Program assists students to settle into their new environment. The college hosts a Year 7 Parent Day in Term 1 where parents have the opportunity to participate in a range of classes with their children. This is followed by a BBQ for parents, students and teachers. New students in other year levels are given a 'buddy' to help them settle in for the first few days of their attendance.
The Student Wellbeing Coordinator is responsible for running school programs such as Peer Support and organising visiting professionals. The Coordinator also deals with a myriad of issues and can organise testing and support for students through outside agencies when appropriate. Our visiting professionals include a speech pathologist, a psychologist and Youth Workers.
1. To develop co-operative and supportive behaviour between all school members including students, parents and teachers.
2. To develop skills and interests in students that will enable them to lead healthy, interesting and productive lives.
3. To identify and assist students who are at risk of leaving school without the opportunity of employment or further education.
Procedures for dealing with harassment and bullying.
Copies of these policies can be obtained by contacting the school and are also in the Student Planner which is distributed to all students.
A School Wellbeing Committee and the Year Level Coordinators meet on a regular basis. Parents are asked to contact their Form Teacher, Year Level or Student Wellbeing Coordinator if any changes occur at home that may have an impact on their son/daughter's learning. Such examples would be family illness or death, change of address, change in family living arrangements or access arrangements, etc.
Our approach to school discipline is based on positive relationships and staff work closely with students to develop a productive and healthy association. The rights and responsibilities of students and staff are negotiated on a regular basis and provide a framework within which all at the college can learn and work productively. All teachers are responsible for maintaining orderly behaviour in and out of class. Our aim is to engage students by providing a comprehensive, and relevant, curriculum and a supportive wellbeing network, combined with high quality teaching. Students are made aware of the expectations of appropriate behaviour and positive participation. The Student Code of Conduct, Harassment Policy and School Rules clearly inform students of their rights and responsibilities, and the consequences if students misbehave. Serious discipline issues are referred to the Level Coordinator, Assistant Principal and/or Principal. Parents are then contacted about these issues.
In most cases, the first person to contact if there are any concerns, is the Form Teacher or Year Level Coordinator. However, parents are most welcome to contact other teachers as appropriate.
A form assembly is held at the start of every school day, form teachers are well placed to assist students. They mark the roll and monitor absence notes, read and discuss the daily bulletin, and assist Student Forum Representatives with various matters. They are also able to deal with any issues or concerns students have.
Year Level Coordinators are responsible for the everyday management of the appropriate year level. They deal with student management issues that may arise which interfere with the productive learning of other students. The Year Level Coordinator works closely with Form Teachers and the Student Wellbeing Coordinator to ensure the best outcome for the student. They will also contact home if the need arises and welcome feedback from families regarding their children.
Where a student qualifies for funding, we will employ an Integration Aide to support the student's learning. However, all staff are committed to developing strategies to enable integration students to participate in classes with their peers.
This program is designed to assist our new Year 7 students with their transition to secondary school and forms a vital role as part of their induction into our College. Year 7 students are placed in small groups and each of these groups is allotted two or three Peer Leaders from Year 10. These Peer Leaders have been trained to lead activities and discussions. This program aims to develop self-esteem, self-acceptance, confidence, the ability to make sound decisions and leadership skills in all students. Peer Support groups meet during the year, starting out quite frequently and tapering off towards the end of the year when the Year 7 students are settled and more comfortable in the school.


text:
'use strict';

const HttpStatus = require('http-status-codes');
const nodemailer = require('nodemailer');
const utils = require('../utils.js');
const usersMngr = require('../bl/usersMngr.js');
const usersPendingMngr = require('../bl/usersPendingMngr.js');
const registerByOnlyEmailDTOs = require('../dto/RegisterByOnlyEmailDTO.js');
const mainModule = require('../app.js');

// create reusable transporter object using the default SMTP transport
let transporter = null;

async function SendConfirmationEmailByOnlyEmail(email, culture, confirmationToken) {
  // create reusable transporter object using the default SMTP transport
  if (!transporter) {
    transporter = nodemailer.createTransport({
      host: process.env.SMTP_HOST,
      port: process.env.SMTP_PORT,
      secure: (process.env.SMTP_SECURE === 'true'), // secure:true for port 465, secure:false for port 587
      requireTLS: true,
      tls: {
        rejectUnauthorized: false,
      },
      auth: {
        user: process.env.SMTP_USER,
        pass: process.env.SMTP_USER_PASSWORD,
      },
    });
  }

  const ConfirmationByOnlyEmailUrl = `${process.env.CONFIRMATION_URL_BY_ONLY_EMAIL}?ct=${confirmationToken}`;

  let subject = process.env[`CONFIRMATION_EMAIL_SUBJECT_${culture}`];
  if (!subject) {
    subject = process.env.CONFIRMATION_EMAIL_SUBJECT_DEFAULT;
  }

  const html = await new Promise((resolve, reject) => {
    mainModule.app.render(`confirmByOnlyEmail-${culture}`, {
      title: process.env.APPLICATION_NAME,
      confirmByOnlyEmailUrl : ConfirmationByOnlyEmailUrl,
      email : process.env.SUPPORT_EMAIL,
    }, (err, renderedHtml) => {
      if (err) {
        reject(err);
        return;
      }
      resolve(renderedHtml);
    });
  });

// setup email data with unicode symbols
  const mailOptions = {
    from: process.env.CONFIRMATION_EMAIL_FROM, // sender address
    to: email, // list of receivers
    subject, // Subject line
    text: ConfirmationByOnlyEmailUrl, // plain text body
    html, // html body
  };

  return new Promise((resolve, reject) => {
    // send mail with defined transport object
    transporter.sendMail(mailOptions, (error, info) => {
      if (error) {
        reject(error);
        return error;
      }
      resolve(info);
      return info;
    });
  });
}

/**
 * Render the index.ejs or index-with-code.js depending on if query param has code or not
 * @param   {Object} req - The request
 * @param   {Object} res - The response
 * @param   {Object} next - The next
 * @returns {undefined}
 */
exports.registerByOnlyEmail = async (req, res, next) => {
  try {
    req.assert('email', 'Email is not valid').isEmail();
    req.sanitize('email').normalizeEmail({ gmail_remove_dots: false });

    try {
      const result = await req.getValidationResult();
      result.throw();
    } catch (e) {
      throw (new utils.ErrorCustom(HttpStatus.BAD_REQUEST, e.mapped(), 4098));
    }

    const email = req.params.email;
    const culture = req.params.culture;

    const user = await usersMngr.findUserByUsername(email);
    if (user) {
      // The client can redirect user to login page and recovery password
      res.json(new registerByOnlyEmailDTOs.RegisterByOnlyEmailDTO(email, '', registerByOnlyEmailDTOs.RegisterByOnlyEmailStatus.UserAlreadyRegistered));
      return;
    }

    let userPending = await usersPendingMngr.findUserByEmail(email);
    if (userPending) {
      // TODO: It may happen that the email is not sent for some problem, but the pending mail
      // has been created,
      // If this is the case, you must notify the Group of Administrators

      await SendConfirmationEmailByOnlyEmail(email, culture, userPending.confirmationToken);

      // The user had tried to register, but for some reason did not confirm,
      // he attempted to enter the email and sent the confirmation email again
      // The token is sent by email, no need to return to the DTO for security issues
      // The client (for example, TiTaggoPublicPortal) should postpone the User
      // to a page informing the User that he should receive an email to confirm the account
      // The e-mail link should point to the client and submit a registration page
      // with Name = "No Name", Surname = "No Surname", email / username =
      // and the Password field to be entered
      res.json(new registerByOnlyEmailDTOs.RegisterByOnlyEmailDTO(email, '', registerByOnlyEmailDTOs.RegisterByOnlyEmailStatus.ConfirmPendingEmailSent));
      return;
    }

    userPending = await usersPendingMngr.createUserPending(email);

    // TODO: It may happen that the email is not sent for some problem, but the pending mail
    // has been created,
    // If this is the case, you must notify the Group of Administrators

    await SendConfirmationEmailByOnlyEmail(email, culture, userPending.confirmationToken);

    // The token is sent by email, no need to return to the DTO for security issues
    // The client (for example, TiTaggoPublicPortal) should postpone the User
    // to a page informing the User that he should receive an email to confirm the account
    // The e-mail link should point to the client and submit a registration page
    // with Name = "No Name", Surname = "No Surname", email / username =
    // and the Password field to be entered
    res.json(new registerByOnlyEmailDTOs.RegisterByOnlyEmailDTO(email, '', registerByOnlyEmailDTOs.RegisterByOnlyEmailStatus.ConfirmPendingEmailSent));
  }  catch (e)  {
    if (e instanceof utils.ErrorCustom) {
      next(e);
      return;
    }
    next(new utils.ErrorCustom(HttpStatus.INTERNAL_SERVER_ERROR, e.message, 4099));
  }
};

exports.confirmAccountByOnlyEmail = async (req, res, next) => {
  try {
    req.checkBody('email', 'Email is not valid').isEmail();
    req.checkBody('name', 'Name is not valid').notEmpty();
    req.checkBody('password', `Password must be at least ${process.env.PASSWORD_MIN_LEN} characters long`).len(process.env.PASSWORD_MIN_LEN);
    req.checkBody('confirmPassword', 'Passwords do not match').equals(req.body.password);
    req.checkBody('confirmationToken', 'Confirmation Token not valid').notEmpty();

    req.sanitize('email').normalizeEmail({ gmail_remove_dots: false });

    try {
      const result = await req.getValidationResult();
      result.throw();
    } catch (e) {
      throw (new utils.ErrorCustom(HttpStatus.BAD_REQUEST, e.mapped(), 4101));
    }

    const email = req.body.email;
    const name = req.body.name;
    const password = req.body.password;
    const confirmationToken = req.body.confirmationToken;

    let user = await usersMngr.findUserByUsername(email);
    if (user) {
      throw new utils.ErrorCustom(HttpStatus.UNAUTHORIZED, 'User registration already exists', 9000);
    }

    const userPending = await usersPendingMngr.findUserByEmailAndConfirmToken(email,
      confirmationToken);
    if (!userPending) {
      throw new utils.ErrorCustom(HttpStatus.NOT_FOUND, 'User pending registration does not exist', 4102);
    }

    user = await usersMngr.createUser(email, name, password, userPending.registrationDate, true,
      confirmationToken, Date.now());

    // TODO: If "usersPendingMngr.deleteUser" fails, a pending User remains in usersMngr
    await usersPendingMngr.deleteUser(userPending);

    res.json(utils.MappingModelAndDTO.toUserDTO(user, false));
  }  catch (e) {
    if (e instanceof utils.ErrorCustom) {
      next(e);
      return;
    }
    next(new utils.ErrorCustom(HttpStatus.INTERNAL_SERVER_ERROR, e.message, 4100));
  }
};

redpajama_set_name: RedPajamaGithub


text:
One Year After The Great East Japan Earthquake (March 11, 2011)
On March 11, 2011, a magnitude 9.0 earthquake occurred off the Pacific coast of Tohoku in Japan. One year later, Japan rebuilds its towns, schools, fisheries, and… hearts.
AsianScientist (Mar. 9, 2012) – On March 11, 2011, a magnitude 9.0 earthquake occurred off the Pacific coast of Tohoku in Japan.
The 2011 Tōhoku-Oki earthquake was the most powerful known earthquake ever to have hit Japan, and it triggered a powerful tsunami and sparked a nuclear crisis and health scare.
NASA researchers later discovered that the destructive tsunami generated by the earthquake was a long-hypothesized "merging tsunami" that doubled in intensity over rugged ocean ridges, amplifying its destructive power before reaching shore.
Satellites captured not just one wave front that day, but at least two, which merged to form a single double-high wave far out at sea – one capable of traveling long distances without losing its power.
At the one year anniversary of the Great East Japan Earthquake, together with international and local support, Japan is slowly rebuilding its towns, schools, and fisheries through acts of resilience and sheer courage.
Resilience: Sunflowers to decontaminate toxic soil
The Japan earthquake and tsunami did more than just cause destruction to houses and roads; it caused severe structural damage to the Fukushima Dai-ichi nuclear power plant, knocking out its water cooling systems and melting the fuel rods inside several reactors, sparking explosions.
In response, Japan raised the crisis level at the beleaguered plant from 5 to 7 – the highest level on the International Atomic Energy Agency's scale for rating nuclear accidents. Food and water safety was also a concern, aside from the health implications of exposure to ionizing radiation.
Shortly after the crisis, a group from the Japan Aerospace Exploration Agency (JAXA), led by Prof. Masamichi Yamashita, grew sunflowers around the nuclear plant to remove soil contamination of cesium.
Radioactive cesium is similar to kalium, a commonly used fertilizer. If kalium is not present, sunflowers will absorb cesium instead. This move was similar to after the 1986 Chernobyl nuclear disaster, where sunflowers and rape blossoms were used to decontaminate soil in Ukraine.
Starting with 300 kilograms of sunflower seeds, the group has sown sunflower seeds in the 30-kilometer region around the Fukushima plant.
Aside from their task of removing radioactive cesium from the soil, the team also hoped that their sunflowers would become a symbol of recovery in the areas affected by the nuclear crisis.
Recovery: UNESCO's Kizuna Campaign
UNESCO's Kizuna campaign, which led to 30,000 messages of friendship from students around the world being sent to their peers in the devastated areas, has brightened the faces of students at Nakano School in Sendai.
During a visit to the UNESCO Associated School Network member school on February 14 last month, Director-General Irina Bokova witnessed both the destruction and the recovery underway.
"I lived in a shelter for four months with my family," said Rikko, whose school has been relocated since the tsunami. "I was very uncertain about the future. Many people came to encourage us. UNESCO and the world were there to support us when we thought we had lost everything."
Teachers are also grateful for the outpouring of support from students across the world.
"Smiles are coming back on the children's faces and we are grateful to UNESCO for its support and solidarity," said the school principal Mr. Mitsuru Takahashi.
"After the earthquake, we had to postpone environmental studies. We started a program of mental care to study in the classroom. Our goal is to make every student able to live their daily school life in a stable condition and sound environment."
At Tohoku University, one of Japan's leading institutions, research equipment, libraries and utilities were damaged, but the university also turned adversity into solidarity.
"We are setting up a new international research institute on disaster science to conduct outcome-oriented research, share experiences internationally and promote cooperation in disaster science," said the University's Vice president Mr. Yukihisa Kitamura.
Return to normalcy: The coastal town of Minami Sanriku
The town of Minami Sanriku, Miyagi Prefecture (Photo: Jerry Velasquez/UNISDR).
Almost one year ago, Minami Sanriku was one of many coastal towns in the Northern Miyagi Prefecture that was devastated by the Tōhoku-Oki earthquake and tsunami. Of its 17,666 residents, some 800 people died after 15-meter tsunami waves pounded the coastal areas.
Today, the town is trying to get back its life, thanks to the help of the 50,000 volunteers and hundreds of residents who, together with government agencies, have worked day and night to clear damage caused by the disaster.
"The double disaster was a real tragedy for all of us," said, Shuji Kounosu, a local resident. "But it has had an incredible positive effect on all of us as it revived the "koh" tradition (the tradition of helping each other), which was previously anchored in the Japanese traditional way of living. This has helped us so much to regain confidence in the future."
And as of last month, a group of 20 universities in Tokyo, including Taisho University and Rikkyo University, have pledged to launch the Minami Sanriku Volunteers Network for the Revival of Tohoku, which will help rebuild local industries, promote tourism, and provide farming assistance.
Fishermen who lost their fishing boats and homes have formed a cooperative in Minami Sanriku near Shizukawa fishing port, and Miyagi Governer Yoshihiro Murai has called for the establishment of a "special reconstruction zone for fisheries."
Prof. Takashi Onishi of the University of Tokyo and President of the Science Council of Japan, has recommended using a community-focused public-private-partnership called 'Machizukuri Corporation' to rebuild local services such as health care and welfare, schools, and energy supplies.
"You can have all the walls you want to protect you but what makes a big difference are the community spirit and the solidarity between people," Kounosu added.
Source: UNESCO; UNISDR; NASA; JAXA.
#Fukushima Dai-Ichi #International Atomic Energy Agency #Japan #Japan Earthquake #JAXA #NASA #Rikkyo University #Taisho University #TEPCO #Tsunami #UNESCO #United Nations International Strategy for Disaster Reduction
300 Tons Of Radioactive Water Leak From TEPCO's Fukushima Plant Japan's nuclear regulators have raised the severity level of a radioactive water leak at the Fukushima plant from one to three on an international scale.
Yukiya Amano Reelected As UN Nuclear Agency Chief Mr. Yukiya Amano has been reappointed to a second term as Director General of the International Atomic Energy Agency.
Half Of Japanese Town Suffers From PTSD Researchers found that eating right, exercising and going to work helped people in the town of Hirono to cope in the aftermath of the Fukushima disaster.
Fukushima Radiation Poses No Immediate Health Risks: UN Science Panel Radiation exposure from the Fukushima nuclear accident is unlikely to result in future health problems, according to a new scientific report by the UN.
Google Maps Shows Japanese Town Two Years After Fukushima Google Street View has released images of a Fukushima town that is facing a grim future two years after the March 2011 earthquake, tsunami, and nuclear meltdown.
Ripples From Japan Earthquake Felt In Space Researchers say that the 9.0 magnitude Japan earthquake of March 2011 was so strong that soundwaves were picked up directly in space.


text:
Q: Failed to compile. static/css/main.b6d8a2cb.css from Css Minimizer plugin TypeError: Cannot set properties of undefined (setting 'value') PS C:\Users\Hp\Desktop\web technology\NCT\carrental> npm run build

carrental@0.1.0 build
react-scripts build

Creating an optimized production build...
Failed to compile.
static/css/main.b6d8a2cb.css from Css Minimizer plugin
TypeError: Cannot set properties of undefined (setting 'value')
the react app runs fine on the development server but shows this error when I try to build it for production ready. crosses checked 100 times, nothing seems to work. the static/css/main.b6d8a2cb.css isn't part of my project.

A: Check all the custom css file (including app.css and index.css) that you have added in your project and look for any missing comments. I had used a custom css file where i had forgot to comment out a few lines properly.


I've read that they displayed their riches by exhibiting and using gold plates and goblets set with precious stones,and also sewing pearls etc. into their garments. The explanation given for this was that there were no banks. Were goldsmiths not yet common? When did the practice of putting ones money with a goldsmith and thus accruing interest begin?
As regular readers will know, I like to celebrate numerical milestones, so I thought I would mention that we've reached 1500 posts! Thanks once again to all the great readers and commenters that help make this blog work. I know I don't say it often enough, but I really appreciate everyone who contributes here!
hello! i'm a fourth year Algerian English student at the university, I am starting to think about my dissertation for this year and am planning to do something on the Tudors Dynasty, I thought of Henry VIII particularly, but I am afraid it is so vague. I studied a little The Tudors Dynasty and watched the series "The Tudors" and I am fan of it and so interested. I would really appreciate any kind of help especially with the outline and content.
If anyone has any suggestions that would narrow down my topic, these would be greatly appreciated.
Wikipedia article on "Battle of Spearhead"
This article might be deleted because it's so short. Can anyone help add to it? I'd be grateful for anything people can add.
How would two dukes address one another? Two kings? Two marquesses? Two earls? Two barons?
There's an article "Hoxun Court" on Wikipedia that may be deleted for lack of information. Can anyone help?
I just finished reading a book about Anne Boleyn. In the book, her sister Mary was the family favorite while Anne was hated. Is this true?
Did noblewomen during the Tudor period customarily keep their maiden names? For example: Margaret Beaufort is never called Margaret Tudor or Margaret Stanley. Lady Jane Grey is never referred to as Lady Jane Dudley. At first, I thought it might have been due to historians seeking clarification; but a portrait of Katherine Parr (artist unknown) shows her with the name Katharine Parre picked out in gold. This portrait does not look to be that of a young girl, which Katherine certainly was when she married for the first time. Any thoughts on this ?
Are there any good books on Jane Seymour? I would prefer fiction, but any suggestions would be appreciated.


text:
Eleanor Grey QC
Silk:
eleanor.grey@39essex.com
Home // Barristers // Eleanor Grey QC
"She is incredibly meticulous, judicious and direct in her dealings. She's also intellectually very powerful."
Chambers and Partners
Eleanor Grey QC took silk in 2011. She has a thorough understanding of public authorities, especially the health and social care sector, and of disciplinary and regulatory proceedings. Clients include the Association of Chartered Certified Accountants, the Department of Health, the Professional Standards Authority and the General Medical Council, as well as numerous other public bodies. As a long-standing member of the Attorney General's 'A' Panel prior to taking silk, she possesses great experience in the field of administrative and public law, including in areas such as mental health. She has also acted in cases involving vulnerable adults in the Court of Protection.
Eleanor has extensive experience of inquiries, reviews and investigations. She has acted as senior Counsel in a number of major public inquiries, most recently the Mid-Staffordshire NHS Foundation Trust Public Inquiry. She is currently acting for one of the core participants in the Independent Inquiry into Child Sexual Abuse. In the last two years, she has chaired an Independent Review of paediatric cardiac services in Bristol and a review of the Disclosure and Barring Service (DBS)'s barring function. She has represented clients in high-profile and sensitive hearings such as the 8-day public hearing of Ian Brady's application to the Mental Health Tribunal for discharge from hospital. She has developed extensive expertise in the law relating to freedom of information and data protection.
Eleanor sits as a Tribunal Judge in the First-Tier Tribunal (Mental Health). In 2014, she was also appointed as a part-time Judge of the Upper Tribunal (Administrative Appeals Chamber).
Eleanor has extensive experience of complex judicial review applications, having spent 8 years on the Attorney-General's "A" Panel (Public Law) before taking silk. Cases include a couple's challenge to the initiation of a child protection investigation into their daughter's wellbeing, a high-profile challenge to the detention of a man on hunger-strike, the British Pregnancy Advisory Service's attempt to overthrow the Department of Health's interpretation of the Abortion Act, and a challenge to the legality of the Department of Health's compensation scheme for Hepatitis C sufferers. She appears regularly in Tribunals, including the Upper Tribunal (where she acted for the Secretary of State for the Home Department in two lengthy 'country guidance' cases on the return of asylum seekers to Zimbabwe) and the First-tier-Tribunal (the Regulatory Chamber and the Health, Education and Social Care Chamber). She has conducted hearings before the Mental Health Tribunal for hospital authorities or for the Secretary of State for Health in Broadmoor, Rampton and Ashworth Special Hospitals. Her work for the Ashworth Hospital authority includes the 8-day public hearing of Ian Brady's application for discharge from hospital, when she cross-examined Ian Brady in his first public appearance since his trial in 1966. According to the BBC coverage: 'Despite … trying to command the room, he did not like being put on the back foot during cross-examination.'
"She is extraordinarily good." Chambers & Partners
Instructed to represent one of the core participants in the Independent Inquiry into Child Sexual Abuse (the Jay Inquiry);
Chair of the Independent Review commissioned by NHS England to examine paediatric cardiac services at the University Hospitals Bristol NHS Foundation Trust (2014 – 2016);
Chair of a Review commissioned by the Disclosure and Barring Service (DBS) of its barring functions, looking at the effectiveness of the DBS's processes for protecting vulnerable children and adults;
Counsel to the Healthcare Commission, Mid-Staffordshire NHS Foundation Trust Public Inquiry (2010 – 2012);
Chair of two working parties for the General Medical Council (GMC), regarding case management and the drafting of charges (2008);
Legal Assistant to Sir Peter Scott QC in his independent investigation into the Crown's conduct of a Northern Irish judicial review, R(Downes) v Secretary of State for Northern Ireland (2007);
Counsel to the Ayling and Kerr/Haslam Inquiries (2002 – 2004): independent inquiries established by the Secretary of State for Health to investigate failures on the part of the NHS to respond to patient concerns about doctors' conduct;
Junior Counsel for the House of Commons Authorities in the Hutton Inquiry (2003);
Junior Counsel to the Bristol Royal Infirmary Inquiry (1999 – 2000), a major healthcare public inquiry;
Has advised interested parties in numerous other inquiries, including the Soham (Bichard) Inquiry.
"A super-steady pair of hands, she's measured, calm and really good." Chambers & Partners
Clients include central government departments, local authorities, healthcare trusts and the House of Commons Authorities. She represented the House of Commons throughout the MPs' Expenses cases. She acted for local authorities in a case concerning fees and charges under the Environmental Information Regulations which established that local authorities had the right to recover the costs of staff time. She has appeared extensively before the Information Rights Tribunal in matters relating to freedom of information and data protection, including on issues such as the confidentiality attached to statements taken for the purpose of investigating serious untoward incidents. She has advised regulators upon matters such as their publication schemes and in the handling of personal information.
"She is extremely thorough and has good judgement." Chambers & Partners
Eleanor acts for regulators such as the Professional Standards Authority, the General Medical Council, the General Dental Council and the Royal Pharmaceutical Society in relation to disciplinary matters. She often represents regulators in appeals to the High Court and Court of Appeal, as well appearing at first-instance disciplinary hearings. She has represented former Primary Care Trusts in hearings before the Family Health Services Appeal Unit (FHSAU). She has acted for pharmaceutical managers in 'control of entry' cases concerning the right to set up new pharmacies. Her advisory practice includes advice on issues such as the division between health and social care. She has acted as a legal advisor to bodies such as the Association of Chartered Certified Accountants, the General Social Care Council and the British Psychoanalytical Council. She chaired two Working Parties for the GMC in 2008, looking at its case management procedures. She is trusted to handle high-profile cases, representing the hospital doctors in the very public Neon Roberts brain cancer treatment case of late 2012, as well as Ian Brady's application for discharge from hospital.
Eleanor is rated in the categories of Administrative and Public Law (Public Inquiries), Data Protection, Immigration, IT and Telecoms and Professional Discipline by Chambers and Partners and the Legal 500.
"A hugely impressive advocate and a delight to deal with." (Chambers & Partners 2017)
"Recommended for health and social care matters" (Legal 500, 2016)
"She has an immense knowledge of the law and a great presence in court." (Legal 500,2015)
"A great presence in court" (Legal 500, 2015)
"A persuasive and formidable advocate." (Legal 500, 2015)
"A super-steady pair of hands, she's measured, calm and really good." "She combines good technical knowledge with a level-headed style of advocacy. She impresses the court by being sensible, pragmatic, balanced, rational and reasonable."(Chambers UK, 2015)
"She is extraordinarily good.""She is extremely thorough and has good judgement."(Chambers UK, 2015)
'A wealth of experience, which is reflected in her focused and measured advice.' (Legal 500, 2014)
'Fantastic, stand-out performer.' (Legal 500, 2014)
'She is incredibly meticulous, judicious and direct in her dealings. She's also intellectually very powerful.'(Chambers UK 2014)
'She was a super opponent in terms of the clarity of her submissions and her fair approach to the case.'(Chambers UK 2014)
'She is superb. Just brilliant.' (Chambers UK 2014)
'I like Eleanor very much; she does great work.' (Chambers UK 2014)
'Fellow practitioners highlight her for her leadership qualities and team working, explaining that she is 'very professional, very knowledgeable, helpful and hard-working.' (Chambers UK 2013)
'A 'superb advocate and extremely effective cross-examiner.' She wins plaudits for her engaging and down-to-earth manner, and is further noted for her excellent analytical skills.' (Chambers UK 2013)
'demonstrates a wide knowledge on information rights' (Legal 500, 2013)
'a wealth of experience' (Legal 500, 2013)
'a natural born advocate' who has 'good judgment and offers spot-on analysis of legal problems' (Chambers UK 2012)
'clear-thinking and thinking and easy to deal with' (Chambers UK 2012)
'an excellent pleader, who is particularly effective at judicial review' (Chambers UK 2012)
'a woman of remarkable intellectual credentials' (Chambers UK 2011)
'skilled practitioner when it comes to public inquiries' (Chambers UK 2010)
Clients report that they are 'impressed with how concise she is and completely calm in all situations.' (Chambers UK 2010)
'a very good style for representing a public authority: very careful, measured and fair-minded' (Legal 500 2009)
'balanced approach' (Chambers UK 2009)
'polished performer' (Chambers UK 2009)
'thorough and understated as an advocate' (Chambers UK 2009)
'measured and amazingly thorough… [her] subtlety belies her effectiveness' (Chambers UK 2007)
'darned good at what she does' (Chambers UK 2007)
'excellent… [impresses with her] thoroughness and her acuity' (Chambers UK 2006)
R (Sailesh Patel) v the General Medical Council (CA) [2013] 1 W.L.R. 2801
Legitimate expectations and alterations to the qualifications needed by overseas students.
R(AB) v London Borough of Haringey [2013] EWHC 416 (Admin); [2013] Fam. Law 965
The conditions required before a lawful child protection investigation could commence.
R. (on the application of Tosar) v Milton Keynes Primary Care Trust [2012] EWHC 2334 (Admin)
Interpretation of the word 'neighbourhood' in the Pharmaceutical Services Regulations.
Flasz & Ors (Claimants) v Havering Primary Care Trust (Defendant) & Secretary of State for Health (Interested Party) : K Jan-Mohamed (Claimant) v Greenwich PCT (Defendant) [2011] EWHC 1487 (Admin)
Representing a primary care trust in a challenge to the means by which contracts for personal medical services could be terminated by a PCT.
British Pregnancy Advisory Service v Secretary of State for Health [2011] EWHC 235 (Admin)
Representing the Secretary of State for Health in a challenge concerning the proper interpretation of the Abortion Act 1967
R (Moore) v Skipton Fund Ltd and Secretary of State For Health [2010] EWHC 3070 (Admin) (201o) 117 BMLR 185
Challenge to legality of Hepatitis C compensation scheme, successfully defended for Secretary of State for Health
R (JB) v (1) A Haddock (2) J Rigby (3) Wood (CA) [2006] EWCA Civ 961
Legality of forcible treatment with anti-psychotics to non consenting patient under the MHA.
R (Rogers) v Swindon NHS Primary Care Trust and Secretary of State for Health (CA) [2006] EWCA Civ 392; [2006] 1 WLR 2649
The availability of Herceptin to women suffering breast cancer.
R (Mersey Care NHS Trust) v Mental Health Tribunal and Ian Stuart Brady (QBD) [2004] EWHC 1749 (Admin); [2005] 1 WLR 2469; [2005] 2 All ER 820; [2005] 83 BMLR 1
An early decision on whether a Tribunal hearing concerning the hospital detention of Ian Brady should be heard in public.
Pam Smith v NE Derbyshire PCT and Secretary of State for Health (QBD) [2006] EWCA Civ 392
The first case to consider the duties of a health authority to consult patients and the public under section 11 of the Health & Social Care Act 2001.
R (KB and Others) v Mental Health Tribunal (QBD) 23 April (2002) ACD 85
Commonplace delays in hearings before the Mental Health Review Tribunal breached the patients' rights to speedy hearings under Art.5(4) European Convention on Human Rights.
Inayatullah v General Medical Council [2015] A.C.D. 34 (charges of dishonesty in Fitness to Practise proceedings).
Professional Standards Authority v General Medical Council [2014] EWHC 1903 (Admin); [2014] Med. L.R. 311 ("online" psychiatric practice and duties owed to patients).
R(Sharaf) v General Medical Council [2013] EWHC 3332 (Admin)
Inappropriate nature of a challenge to part-heard fitness to practice proceedings.
Michalak v General Medical Council [2011] EWHC 2307 (Admin)
The GMC had the power to appoint substitute Panel Members to a Fitness Practise Panel when a member was unable to continue sitting.
Yeong v General Medical Council (2009) 110 BMLR 125, (2010) 1 WLR 548
The limits of the importance of remediation when judging current impairment to practice.
Council for the Regulation of Healthcare Professionals v (1) General Medical Council (2) Tarun Kumar Biswas (QBD) 26 January 2006 [2006] EWHC 464 (Admin)
Appeal against GMC decision by the Council for the Regulation of Healthcare Professions.
Giele (Henricus) v General Medical Council (QBD) 14 October 2005 [2005] EWHC 2143 (Admin); [2006] 1 WLR 942; [2005] 4 All ER 1242; [2006] 87 BMLR 34
The sanction to be applied following an improper sexual relationship between Dr G and a patient.
Sengupta v General Medical Council and others (CA) 31 July 2002 [2002] EWCA Civ 1104
Circumstances when a judge should recuse himself for bias (instructed as amicus by the Attorney-General).
R(Muaza) v Secretary of State for the Home Department (CA) [2014] 1 WLR 1870
The legality of detaining a person at risk of death through hunger-strikes.
R(OM by her Litigation Friend the Official Solicitor) v Secretary of State for the Home Department (CA) [2011] EWCA Civ 909
A complex false imprisonment case concerning the detention of the mentally ill, and the rights of a child.
R (Dbeis) v Secretary of State for the Home Department (CA) 19 May 2005 [2005] EWCA Civ 584; The Times 30 May 2005
The removal of a child to the Lebanon was not contrary to Article 8, ECHR.
JM v Secretary of State for the Home Department (CA) 4 October 2006 [2006] EWCA Civ 1402
The circumstances in which a human rights appeal, as well as a refugee appeal, may be made by a claimant.
R v Ashworth Special Hospital Authority, ex parte E (QBD) 19 December 2001 [2001] EWHC Admin 1089; The Times 17 January 2002
This was an application for judicial review of a decision of the Defendant special hospital only to allow the Claimant, a male patient, to wear women's underwear in his own room.
Leeds City Council v IC EA/2012/0020-0021
Entitlement to levy fees for staff time answering a request under the Environmental Information Regulations 2004.
JW v IC (FTT)
Protection of medical records belonging to a deceased patient.
Roberts v IC and Dyfed Powys Police Authority EA/2012/0032
Interrelationship between subject access requests under s7, DPA and s40(2) of the Freedom of Information Act 2000.
Eleanor's Practice Areas:
Eleanor's Practice Areas
Eleanor's Sectors
Infected Blood Inquiry Formally Opens
Infected Blood Inquiry: Representation of Core Participants
Capacity, limitation and the Human Rights Act (AP (by his litigation friend, BA) v Tameside Metropolitan Borough Council (Eleanor Grey QC – Lexis PSL)
The Bristol Review publishes final report into children's cardiac services
Eleanor Grey QC representing Ashworth Hospital in Ian Brady's Mental Health Tribunal
39 Essex Chambers to speak and exhibit at the LLG Governance Conference 2018 in Gateshead
Local Government Conference 2018
Eleanor Grey QC to be guest speaker at Clyde & Co seminar "GDPR In Healthcare – Are You Ready?"
Upholding and Challenging: Handling Regulatory and Disciplinary Appeals
Current Legal Challenges for the NHS
FOIA Update and Consultation Seminar – Cardiff
Local Government Law Update
Recent Developments in Public Law
Articles 2 & 3 of the European Convention on Human Rights: Current themes and recent developments


text:
using Audit.Core;

namespace Audit.WCF
{
    public static class AuditEventExtensions
    {
        /// <summary>
        /// Gets the WCF Event portion of the Audit Event for a given scope.
        /// </summary>
        /// <param name="auditScope">The audit scope.</param>
        public static WcfEvent GetWcfAuditAction(this AuditScope auditScope)
        {
            return auditScope?.Event.GetWcfAuditAction();
        }

        /// <summary>
        /// Gets the WCF Event portion of the Audit Event.
        /// </summary>
        /// <param name="auditEvent">The audit event.</param>
        public static WcfEvent GetWcfAuditAction(this AuditEvent auditEvent)
        {
            if (auditEvent is AuditEventWcfAction)
            {
                return (auditEvent as AuditEventWcfAction).WcfEvent;
            }
            // For backwards compatibility
            return auditEvent.CustomFields.ContainsKey("WcfEvent")
                ? Configuration.JsonAdapter.ToObject<WcfEvent>(auditEvent.CustomFields["WcfEvent"])
                : null;
        }
    }
}


text:
Sahara Group's 2020 sustainable development report emphasizes growth and innovation
By Arnold D. Snapp On Dec 26, 2021
Cover of the Sahara Group's sustainable development report
Sahara Group has published its 2020 sustainable development report which reiterates its commitment to create shared value for its various stakeholders
We have navigated through global economic and health challenges with constant attention to creating shared value for our stakeholders. Our proactive response to the pandemic ensured business continuity "
– Pearl Uzokwe, Director, Governance and Sustainability, Sahara Group
LAGOS, NIGERIA, December 26, 2021 /EINPresswire.com/ –
A leading energy conglomerate, Sahara Group has published its 2020 sustainable development report which also reiterates its commitment to create shared value for its various stakeholders through economic development, environmental protection and the construction of a sustainable society.
The report highlights the economic, social and environmental performance of the Group through 7 operational entities within the Group, including the Sahara Foundation.
Under the theme "Harnessing disruption for growth and innovation", the report highlights how Sahara continues to leverage innovation and technology to achieve its corporate goals and sustainability ambitions in its activities by Africa, Asia, Europe and the Middle East. In 2020, the Sahara Group took on the global economic and health challenges, "leveraging the disruption for growth and innovation".
Sahara Group Governance and Sustainability Director Pearl Uzokwe said the group has continued to foster partnerships and initiatives that have co-created a desirable future through innovation.
Uzokwe said, "We have traversed global economic and health challenges with constant attention to creating shared value for our stakeholders. Our proactive response to the pandemic has ensured business continuity."
She said the Group's COVID-19 response work included launching Abuja's only isolation, recovery and treatment center with 300 patient beds; provide and maintain the oxygen supply to hospitals in Ghana and Côte d'Ivoire, provide dry and cooked food to more than one million beneficiaries, face masks, hand sanitizers and relief supplies to communities where our electricity subsidiaries (Ikeja Electric, Egbin Power and First Independent Power Limited), upstream and other Sahara subsidiaries operate across Africa.
In Zambia, Asharami Energy Limited Company donated tens of thousands of handcrafted fabric / chitenge masks to increase access to PPE and slow the spread of the virus. The company has also published materials in English, Bemba and Nyanja to facilitate the dissemination of information about the pandemic. In Kenya, Asharami Synergy Limited donated thousands of 5 liter jerry cans of hand sanitizer, while oxygen was donated to hospitals in Nigeria (via Egbin Power), Ghana as well as Côte d 'Ivory to stimulate survival operations. Our commitment is to impact 1.3 billion Africans as the continent recovers from the pandemic.
According to Uzokwe, Asharami Energy Limited, an upstream company of the Sahara group, recorded an improvement in responsible sourcing performance with 92% of our suppliers assessed for environmental risks in 2020. A leading operator in the sector in upstream from Africa, Asharami Energy also achieved zero fatalities and losses. time of injury during the reporting year.
Sahara's commercial branch is committed to fully comply with IMO2020 regulations on maintaining the sulfur content in marine fuels at 0.50% m / m (mass by mass) for emission control. In addition, a commitment of $ 44 million for the construction of liquefied petroleum gas (LPG) facilities in the West African region was made as well as the purchase of two additional LPG vessels to ensure the availability of cleaner energy sources for our global market.
"Our commitment to host communities has enabled us to invest US $ 742 million in community development, with a focus on supporting COVID-19 relief efforts. We also repaired all cables and panels for the rural electrification project in the community of Ajoki. A project that we plan to complete by 2022. These demonstrate our progress on the path to sustainable development, "Uzokwe added.
Despite the challenges faced in 2020, the Sahara Foundation has realigned its areas of intervention to address two new pillars: promoting access to clean energy and fostering sustainable environments as we execute the social responsibility initiatives of people and companies. (RPC) of the Group.
In 2020, we launched a flagship project, the Sahara Impact Fund, aimed at strengthening the UNSDGs in Africa with a capacity of over $ 100,000.
Sahara Group's 2020 sustainable development report reflects its economic, social and environmental activities from January 1 to December 31, 2020. The report is the energy conglomerate's sixth sustainable development report and the fifth report drawn up in accordance with the GRI standard. The 2020 sustainability report has been organized and presented in accordance with the sustainability reporting standards of the Global Reporting Initiative (GRI). The guidelines aim to ensure consistency between companies reporting on their sustainable development activities.
For detailed information, please visit the Sahara Group website (www.sahara-group.com) to download the full report
Bethel Obioma
Saharan group
HANATU THE FILM
December 26, 2021, 6:47 p.m. GMT
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text:
FLC Business Search Blast-Off on Mission: SPACE
Blast-Off on Mission: SPACE
https://spinoff.nasa.gov/database/spinoffDetail.php?this=/spinoff//jsc/J...
Mission: SPACE, an attraction at Walt Disney World's Epcot theme park in Orlando, Florida, features cutting-edge ride technology that gives guests the incredible sensation of lifting off and traveling through space on a mission to Mars. With input from current and former NASA advisors, astronauts, and scientists, Walt Disney Imagineering developed the attraction to be the first ride system capable of taking guests straight up into simulated flight for a one-of-a-kind astronaut experience. Passengers experience sensations similar to what astronauts feel during liftoff, as they hear the roar of the engines and view computer-generated, photo-realistic imagery based on data taken from NASA's Mars-orbiting satellites.
Internal Laboratory Ref #:
JSC-SO-219
Patent Status:
NASA Johnson Space Center
Tech Transfer Website
Lab Representatives
Linda Ham
Linda.j.ham@nasa.gov


Please note due to high demand for services, it may take a few days to be able to get back to you.
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