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In late June of 1945, one woman is escorting another through a US checkpoint in Berlin. The young American soldier is somewhat cocky and feigns an officious sort of suspicion that causes him to demand the bandaged woman reveal her face – just to be sure. The obvious agony she feels just beginning to unwind the various layers which hide her identity is enough to convince him that he's made a cruel mistake and he lets the pair pass.
Finally Lene (Nina Kunzendorf) delivers the wounded Nelly (Nina Hoss) to a specialist hospital. A survivor of Auschwitz, Nelly was at some point shot in the head and left for dead. Though she miraculously survived, her face is ruined – missing nose, shattered cheekbones etc. She will need extensive reconstructive surgery. "Who would you like to be?" her doctor asks her, but Nelly only wants to be herself – exactly as she was. The doctor advises against it – it can never be exactly the same and the uncanniness is something not everyone can get over plus it can be an advantage to be given the opportunity to start all over again with a new face, a new identity newly shed of all the scars of a traumatic past. Nelly, however, is insistent.
Returning to the city with Lene, she learns that her entire family and many of their mutual friends have been killed though others turned out to have been nazi sympathisers. Nelly repeatedly asks about her husband, Johnny (Ronald Zehrfeld), but Lene is reluctant to talk about him. Roaming the streets alone at night she tracks him down to a seedy cabaret club, Pheonix, in the American sector where he now clears tables rather than playing the piano. When she calls him by his former name he barely reacts and fails to recognise her. Later he tracks Nelly down and makes her a very odd proposition – pretend to be his deceased wife to claim the inheritance then split the proceeds.
"I no longer exist" exclaims Nelly at one point. Robbed of everything apart from her breath, Nelly has been erased and replaced by something with no clear history. She wants to go back, to reclaim the life she led before exactly how it was but her home no longer exists – her city is in rubble, most of her friends are dead and the husband that she's made the anchor of her survival may have been the very one who betrayed her.
Meeting Johnny (now "Johannes") again and moving into his back room she studies for the role of a lifetime – once again inhabiting her former self, stepping into the shoes of a soulless ghost. Nelly pleads with him silently to remember – recall her from the abyss, recognise her living form as the woman who was taken away in October 1944. Johnny, however, cannot bear to think about the past. He's convinced himself his wife is dead and is only interested in claiming her money to make a new life in the post-war world. No matter how the coincidences mount up as "Esther" not only looks like "Nelly" but also has her handwriting, voice and movement, Johnny refuses to recognise her or acknowledge their shared tragedy.
Operating like an inverted Vertigo, Phoenix is an extremely rich character drama which not only deals with one woman rebuilding herself from the ashes but also with her nation's sense of guilt as it resolutely refuses to look the victims of its crime in the eye. Nelly needs to remember and have her existence acknowledged in order to reclaim her identity, but Johnny cannot bear to look, his guilt is so great that it would shatter his sense of self irrevocably. They dance around each other caught between past and future but both trapped, their passage blocked by the symbolic checkpoints that exist all around them in their now ruined city.
Just as the doctor told her, it can never be exactly the same. At the end of the film, Nelly's transformation is complete, her selfhood restored though somehow different from before. Lene wanted to run away to Palestine, create a new world for her people free from fear and persecution, Johnny wanted to forget and Nelly needed to remember (and be remembered) in order to become herself again but in the end nobody gets quite what they wanted. Only Nelly by meeting her former self head on is able to evolve, finally pulling away from us, out of focus.
Petzold serves us ghosts of several varieties including those of our cinematic pasts by imbuing his melodrama with the gloomy allure of the film noir mixed with the uncomfortable psychology of the Hitchcockian thriller and the uncanny horror of Eyes Without a Face. Probing questions of identity which extend from the individual to the national it asks us to consider a post-war world of guilt and recrimination in which everyone is engaged in rebuilding an idea of selfhood which can take account of wounds suffered or inflicted. Difficult and complex yet beautiful too, Phoenix is anchored by the extremely accomplished performance of its star Nina Hoss and proves a hauntingly melancholy exploration of all it means to be alive.
Phoenix is currently available in the UK on blu-ray, DVD and VOD courtesy of SODA Pictures and is available in the US as part of the Criterion Collection.


Situated at the heart of Fairfield County, Richter Park Golf Course is situated along the West Lake Reservoir. The course has been ranked among not only the top ten golf courses in Connecticut but also as one of the top 25 public courses in the country. The course sprawls across more than 180 acres of gently rolling hills. Water has a prominent presence throughout, with the Reservoir bordering seven holes on the front nine and water hazards coming into play on a total of 14 holes. In addition to the water, there are a variety of elevation changes ranging from subtle to dramatic, which can make it a tough golf course to play your first time. Be prepared for a few blind shots. Richter Park's signature hole, the par-5 12th, is bordered by Boggs Pond. Your third shot will be over water to a bunkered green.
I have always loved this course and bring out-of-town visitors that rave about such a great course being available to the public . It is challenging, scenic and has a welcoming staff but all these things cannot overcome what mother nature has done to holes 10, 11, 14, 15 & 18. I will be back in July to see how things work out.
Always a challenging & interesting round of golf no matter the conditions. Course is still wet from winter rains & many holes have been shortened. Good play for price. Maintenance doing great job to maintain & upgrade because of weather!
Course was in great condition considering weather and repairs to areas damaged due to vandalism. Fantastic Fall vistas from many of the tees. Always provides challenge to be accurate, keep in fairways and greens were fast & great condition. Is always a must play!
The course markets itself as the premier public course in Connecticut and one of the best in the country, and I just don't buy it. The greens were fine, but the rest of the course was in average condition at best. While the layout around the lake on the front nine was very nice, I was disappointed overall, especially given its hefty $60 price tag in the shoulder season ($80 normally). The lack of a driving range also hurt. Wasn't necessarily a bad course, but it just wasn't up to the standard it claims to hold.
Unfortunately the course was in terrible shape. I know that this season has been both humid and rainy but played Ridgefield and Smith last week and both were in tremendous shape.
We also waited at EVERY SINGLE HOLE!
No rangers around to move things along..
Won't be going back anytime soon.
A game of two halves....a slow game.
Front nine was magnificent. A couple of blind holes, but good condition and beautiful landscape. Back nine was just awful. Greens were decent enough, but most fairways were trash and this is with cart path only rule. I get the wet summer, but other courses are holding up much better.
Worse than that - five hours on a Friday is just brutal..letting guys walk the course doesn't help when it is back to back. Super friendly staff, but they've got to get on that speed of play..
Expecting better from a course with this reputation and at this price point.
A very challenging golf course, a good test of golf.
Played here twice both times had a awesome experience . Coarse plays out great greens are perfect and quick great role on your ball . Fairways are perfect price wise on target . Go out and enjoy a great round of golf !
First time I've played here in a few years. The pace of play was excellent, finished in just under 4 hours on a Sunday morning. The greens were perfect, smooth and fast. The rest of the course could use some work, inconsistent bunker sand, some old bunkers were converted to grass bunkers and were ground under repair. Some tee boxes were In rough shape, could be leveled and re sodded. With some TLC, this course could be even better. Overall, good course, just lacking attention to detail to take the next step.
Richter Park is an excellent public golf course. It is well run, good amenities, however play on the weekends can be torturously slow.
I play this course 2-3 time per year, my above ratings of overall (good) is due to pace of play, if not teeing off very early hard to finish in 4 1/2 hours. I understand being customer friendly BUT Pace must be inforced it's hurting game not helping. Plus price course charges a non- Danbury resident is too high for average player , understand giving Danbury residents a great value price wise but to charge players outside city limits in mid-high $80's not using Golf now or tee-off sites is on high side.
I know maintaing conditions is expensive. This is such a great layout would like more people to have access to play.
Top public courses in Fairfield County, Conn.


Plenty of fear and loathing of the nerd rapture backed by powerful VCs in this Harper's article. Ungated version.
Discussed in depth at LessWrong.
... Be explorers; take advantage of this vast new landscape that's been opened up to us in this time and this place; and bear the torch of applied rationality like brave explorers. And then, like, keep in touch by email." The workshop attendees put giant Post-its on the walls expressing the lessons they hoped to take with them. A blue one read RATIONALITY IS SYSTEMATIZED WINNING. Above it, in pink: THERE ARE OTHER PEOPLE WHO THINK LIKE ME. I AM NOT ALONE.
Makes me wish I still lived in the Bay Area :-) My MIRI interview.
I didn't read much loathing in it. A few cheap laughs, maybe.


Jessica turned 5 last month. Now that the term has started, she is officially compulsory school age. Not that that means anything really to us.
– maths: she practises by asking us – and asking us to ask her – addition and subtraction questions, including hundreds, crossing tens and crossing hundreds. She's beginning to multiply, eg she asked Daddy, "What's 10+10+10?", and when he said 20, she said, "No, I said three the tens, not two!" She recognises relationships eg 2+3=5 so 20+30=50 and completed some workbook questions on this. She goes below zero when she asks something like 5 take away 10, and understands how minus numbers extend from zero. She knows half and quarter of shapes, eg toast. She tells the time to o'clock and half past and estimates time intervals well, estimates distances, measures accurately using a ruler, eg twilight sparkle barbie is 9 inches, the mini doll is 4 and a half inches.
– seasons – she keeps an eye on the weather getting warmer and sunnier, reads our outdoor thermometer, eg she read it at 19 degrees by going back one from 20. She spots blossom, flowers, bluebells, butterflies, bees, ladybirds and knows this is because summer is coming. She appreciates and notices that daylight length is longer now, evenings are lighter, the sun is shining when we wake up.
She has also talked about saving up money to go to the toy shop. She knows that she can save some money each week if she only spends part of her pocket money and saves some instead. She's beginning to learn about finances.
It's a busy time in her life right now. So much to learn about. But I think she loves it.
→ Look at me, I am five!


Hospitality and catering vocational courses at Accrington and Rossendale College have been scrapped because of a lack of demand.
College bosses say the department has seen a 'downward trend over the last few years' with more students opting for work-based programmes instead of full-time courses.
Organisers of the Hyndburn Business Award said they have now been 'left with a problem' after the college confirmed they will be unable to provide catering for this year's ceremony.
There are currently 14 students on hospitality and catering courses with 17 students applied to start in the next academic year. However the college expect less than 40pc of the applicants would eventually enrol on the courses.
They are now being offered 'impartial advice and guidance' from the college's careers advisers.
The existing apprenticeship route will not be affected by the changes.
Principal Amanda Melton said: "It is with regret that we are no longer able to continue our hospitality and catering vocational provision.
"Student recruitment in the hospitality and catering department has seen a downward trend over the last few years, not only at ARC, but nationally as well.
"More students are opting for work-based provision, instead of full-time programmes, and as such our apprenticeship catering provision will remain the same.
"To maintain a stable and secure financial position for the college, we need to regularly review the provision we offer to ensure that it remains financially viable.
"A number of factors are considered including student recruitment, retention and destinations in work, as well as the cost of offering the provision such as facilities and staffing.
"The staff affected by these proposals have been briefed and remain a priority to us over the coming weeks, and we will remain committed to helping our current students complete their studies successfully and progress on to positive destinations.
Rob Carder, of Enterprising People, who organise the Hyndburn Business Awards, said the college was expected to provide food for more than 250 guests on June 18 and has appealed for help to find a new caterer.
He said: "For the last three years the meal at the awards' ceremony has been catered for by Accrington and Rossendale College, and a wonderful job they have done of it too.
"Sadly, the catering service is about to be closed, and they won't be able to support the ceremony this year.
To get involved contact rob@enterprisingpeople.org.uk.


AshleyMichelle sat and talked about how therapy earlier in his life would have affected his career, being the "guy in the room", and whether he receives special privileges since he's now light skinned Charlamagne.
We caught up with the New York Times Best Selling author before his book signing at Barnes & Noble in Houston, TX.
"In this inspirational blueprint for working on your mental health, Charlamagne tackles the anxieties that have plagued him his whole life so that he can hopefully keep reaching the next levels of his potential.


text:
\section{Introduction}

The inversion number and major index for the permutation  group are two of the most important statistics
which have received remarkable attention  in the combinatorial
literature, see, e.g., \cite{Foata,FandS,Garsia,GandG,Goulden,MacMahon}. Extensions of these two statistics have been intensely studied. The present paper concerns two directions of such extensions: One is toward labeled forests duo to Bj\"{o}rner and Wachs \cite{BW}, and the other is toward the Coxeter groups (mainly types $B$ and $D$), see, e.g., \cite{ABR,ABR1,AR,Biagioli,Biagioli2,BZ,Chow,Reiner}.

For integers $m,n$ $(m\leq n)$, we use $[m,n]$ to denote the interval $\{m,m+1,\ldots,n\}$.  Denote by $S_n$ the permutation group on $[1,n]$. Note that we use $[1,n]$, instead of $[n]$, to denote the set $\{1, 2, \ldots n\}$
before we shall use $[n]$ to denote the $q$-number $1+q+q^2+\cdots + q^{n-1}$.
We shall
represent a permutation $\pi\in S_n$ in one-line notation
$\pi=\pi_1\cdots\pi_n$. Then the descent set of $\pi$ is defined by
\begin{equation*}
\mathrm{Des}(\pi)=\{i\colon\ 1\leq i\leq n-1,\, \pi_i>\pi_{i+1}\}.
\end{equation*}
The inversion number and major index of $\pi$ are defined by
\begin{align*}
\mathrm{inv}(\pi)&=|\{(i,j)\colon\ 1\leq i<j\leq n,\, \pi_i>\pi_j\}|,\\[5pt]
\mathrm{maj}(\pi)&=\sum_{i\in \mathrm{Des}(\pi)}i.
\end{align*}
The following classical formula is  duo to  MacMahon \cite{MacMahon}
\begin{equation}\label{MacMahon1}
\sum_{\pi \in S_n}q^{\mathrm{inv}(\pi)}=[n]!=\sum_{\pi \in
S_n}q^{\mathrm{maj}(\pi)},
\end{equation} where $[n]=1+q+\cdots+q^{n-1}$, and $[n]!=[1][2]\cdots
[n]$.

The inversion number and the major index on permutations have been generalized
by Bj\"{o}rner and Wachs \cite{BW} to labeled forests.
Let $F$ be a (plane) forest with vertex set $V(F)$. The reason that we consider plane forests is that every vertex can be viewed as having
a unique position in the
 sense that all the vertices are implicitly labeled.
 A labeling $w$ of $F$ is a
bijection
\[w\colon\ V(F)\longrightarrow [1,n].\] For each vertex $u\in F$, the
hook length of $u$, denoted by $h_u$, is the size of
the subtree rooted at $u$. When $F$ is considered  as a
poset with roots at the top, the hook length of $u$ equals the  cardinality of
the principle ideal $\{v\in F\colon\ v\leq_F u\}$ where $\leq_F$ is the order relation.
Bj\"{o}rner and Wachs \cite{BW} defined the descent set of a labeled forest
as given below,
\[
\mathrm{Des}(F,w)=\{u\in F\colon\ w(u)>w(v),\, \mbox{$v$ is the
parent of $u$}\},
\]
and
\begin{align}
\mathrm{inv}(F,w)&=|\{(u,v)\colon\ u>_Fv,\, w(u)<w(v)\}|,\label{BWinv}\\[5pt]
\mathrm{maj}(F,w)&=\sum_{u\in \mathrm{Des}(F,w)}h_u.\label{BWmaj}
\end{align}
If $F$ is a linear tree, then we get a  permutation by reading
 the labels bottom up. The descent set, the inversion number and the major index for
a linear tree  coincide with the corresponding indices
for permutations. Bj\"{o}rner and Wachs \cite{BW} derived the
following $q$-hook length formula by $q$-counting all labelings of a
fixed forest with respect to the inversion number   and the major index, which reduces to the formula (\ref{MacMahon1}) by
restricting $F$ to be a linear tree.

\begin{theorem}[\mdseries{Bj\"{o}rner and Wachs \cite{BW}, Theorem
1.3}]\label{TBJ1} Let $F$ be a forest of size $n$. Then
\begin{equation}\label{BW}
\sum_{(F,w)}q^{\mathrm{inv}(F,w)}=\frac{n!}{\prod_{u\in
F}h_u}\prod_{u\in
F}[h_u]=\sum_{(F,w)}q^{\mathrm{maj}(F,w)},\end{equation} where
$w$ ranges over all labelings of $F$.
\end{theorem}

In this paper, we shall be concerned with signed labelings of a (plane)
forest.
The signed permutation group $B_n$ is  the group of  bijections $\sigma$ on the set $[-n, n]\backslash \{0\}$ such that
\[\sigma(-i)=-\sigma(i)\]
for $i\in [-n, n]\backslash \{0\}$. Recall that $B_n$ is also known as the hyperoctahedral group of rank $n$, or the Coxeter group of type $B_n$. For $\sigma\in B_n$, we write $\sigma$ in the one-line notation $\sigma=\sigma_1\cdots \sigma_n$, where $\sigma_i=\sigma(i)$ for $i\in [1,n]$. In the language of Coxeter groups, $B_n$ is the Coxeter group of type $B_n$ with respect to the generating set
$\{s_0,s_1,\ldots,s_{n-1}\}$, where $s_i$,
$i=1,2,\ldots,n-1$, are the simple transpositions
\[s_i=(1,\dots,i-1,i+1,i,i+2,\ldots,n)\]and $s_0$ is the sign
change\[s_0=(-1,2,\ldots,n).\]

The length function of an element $\pi$ in a Coxeter group, denoted by $\ell(\pi)$, is the minimum number of
generators that occur in its factorizations, see, Bj\"{o}rner and Brenti \cite{BB}, namely,
\[\ell(\pi)=\min\{r\colon\ \pi=s_{i_1}s_{i_2}\cdots s_{i_r},\
\mbox{$s_{i_1},s_{i_2},\ldots, s_{i_r}$ are generarors}\}.\]

For $B_n$, there exists a simple combinatorial interpretation for the length function.  Let $\sigma=\sigma_1\cdots\sigma_n$ be a signed permutation, and define
\begin{align}
\mathrm{n_1}(\sigma)&=|\{\sigma_i\colon\ 1\leq i\leq n,\ \sigma_i<0\}|,\\[5pt]
\mathrm{n_2}(\sigma)&=\left|\big\{\{i,j\}\colon\ 1\leq i,j\leq n,\
\sigma_i+\sigma_j<0\big\}\right|.\label{N2}
\end{align}
Then the length function of $\sigma$ is given by
\begin{equation}\label{LFB}
\ell_B(\sigma)=\mathrm{inv}(\sigma)+\mathrm{n_1}(\sigma)+\mathrm{n_2}(\sigma),
\end{equation}which can be seen as the inversion number for signed permutations, see, Biagioli \cite{Biagioli}, or Bj\"{o}rner and Brenti \cite{BB}.
The following length generating function is well-known.

\begin{theorem}[\mdseries{Humphreys \cite[Section 3.15]{Humphreys}}]\label{LGFB}
\begin{equation}\label{lenB}
\sum_{\sigma\in B_n}q^{\ell_B(\sigma)}=[2][4]\cdots[2n].
\end{equation}
\end{theorem}

As a subgroup of $B_n$, the group of even-signed permutations, i.e., permutations with an even number of minus signs, is  denoted by $D_n$. It is
well known that  $D_n$ is the Coxeter group  respect to the generating set $\{t_0, s_1,\ldots, s_{n-1}\}$, where $s_i$ for $1\leq i\leq n-1$ are defined as above and
\[t_0=(-2,-1,3,\ldots,n).\]
For any signed permutation $\sigma$, let
\[\mathrm{Neg}(\sigma)=\{1\leq i\leq n\colon \sigma_i<0\}.\]
The length function for $D_n$ can be computed by the following combinatorial
formula
\begin{equation}\label{lengthD}
\ell_D(\sigma)=\mathrm{inv}(\sigma)-\sum_{i\in \mathrm{Neg}(\sigma)}\sigma_i-\mathrm{n}_1(\sigma),
\end{equation}which can be considered as the inversion number of an even-signed permutation, see, Bj\"{o}rner and Brenti \cite[Section 8.2]{BB}, or Biagioli \cite{Biagioli2}.
It can be checked that
\[-\sum_{i\in \mathrm{Neg}(\sigma)}\sigma_i=\mathrm{n}_1(\sigma)+\mathrm{n}_2(\sigma).\]
So (\ref{lengthD}) can be reformulated as
\begin{equation}\label{LFD}
\ell_D(\sigma)=\mathrm{inv}(\sigma)+\mathrm{n}_2(\sigma).
\end{equation}

The length generating function for $D_n$ is given by the following formula.

\begin{theorem}[\mdseries{Humphreys \cite[Section 3.15]{Humphreys}}]\label{LGFD}
\begin{equation}\label{lenD1}
\sum_{\sigma\in D_n}q^{\ell_D(\sigma)}=[2][4]\cdots[2n-2][n].
\end{equation}
\end{theorem}

Statistics on Coxeter groups  that are equiditributed with  the length function
are called Mahonian.
 An important statistic on $B_n$ is the flag major index introduced by Adin and Roichman \cite{AR},
which is defined in terms of Coxeter elements and can be expressed combinatorially as
\begin{equation}\label{flagmaj}
\mathrm{fmaj}(\sigma)=2\,\mathrm{maj}(\sigma)+\mathrm{n_1}(\sigma).
\end{equation}

The second Mahonian major statistic on $B_n$ is the negative major index introduced by  Adin,  Brenti and  Roichman \cite{ABR}, which has the following combinatorial description
\begin{equation}\label{nmaj}
\mathrm{nmaj}(\sigma)=\mathrm{maj}(\sigma)+\mathrm{n}_1(\sigma)+\mathrm{n}_2(\sigma).
\end{equation}

There is another Mahonian  statistic which is based on  the major index  defined by Reiner \cite{Reiner}. Under the following order
\begin{equation}\label{order}
1<\cdots<n<-n<\cdots<-1,
\end{equation}
the descent set $\mathrm{Des}_R(\sigma)$ is defined as
\begin{equation*}
\mathrm{Des}_R(\sigma)=\{i\in [1,n]\colon\, \sigma_i>\sigma_{i+1}\},
\end{equation*}
under the assumption that $\sigma_{n+1}=n$.
Then the major index $\mathrm{maj}_R(\sigma)$ is given by
\begin{equation}\label{Rmaj}
\mathrm{maj}_R(\sigma)=\sum_{i\in \mathrm{Des}_R(\sigma)}i.
\end{equation}
Reiner \cite{Reiner} has shown that
\begin{equation}\label{rg}
\sum_{\sigma\in B_n}t^{\mathrm{n_1}(\sigma)}q^{\mathrm{maj}_R(\sigma)}=(1+tq)^n[n]!.
\end{equation}
While the index $\mathrm{maj}_R$ is not Mahonian,
as observed by Biaginoli and Zeng \cite{BZ}, from   (\ref{rg}) it follows that
the index
\begin{equation}\label{zeng}
2\,\mathrm{maj}_R(\sigma)-\mathrm{n}_1(\sigma)
\end{equation} is  equidistributed with the flag major index (\ref{flagmaj}).
They also gave a proof of this fact by justifying the following relation
\begin{equation*}
\mathrm{maj}_R(\sigma)=\mathrm{maj}(\sigma)+\mathrm{n}_1(\sigma).\end{equation*}

We next define a new Mahonian index called R-major index for signed permutations which relies on the major index $\mathrm{maj}_B$ with respect to
 the natural order
 \begin{equation}\label{norder}
  -n < \cdots <-1<0< 1<\cdots < n.
  \end{equation}
The index $\mathrm{maj}_B$  can be shown to be isomorphic to Reiner's major index (\ref{Rmaj}) defined with respect to the order (\ref{order}).
The descent set $\mathrm{Des}_B(\sigma)$ is described in terms of the natural order; that is,
\begin{equation}
\mathrm{Des}_B(\sigma)=\{i\in [1,n]\colon\, \sigma_i>\sigma_{i+1}\},
\end{equation} where $\sigma_{n+1}=0$.  Let
\begin{equation}\label{PRmaj}
\mathrm{maj}_B(\sigma)=\sum_{i\in \mathrm{Des}_B(\sigma)}i,
\end{equation}
and let $\mathrm{p}(\sigma)$ be the number of positive entries of $\sigma$. Then the R-major index is given by
\begin{equation}\label{positive}
\mathrm{rmaj}(\sigma)=2\,\mathrm{maj}_B(\sigma)-\mathrm{p}(\sigma).
\end{equation}

It can be shown that the  index $\mathrm{maj}_B(\sigma)$ in (\ref{PRmaj}) is isomorphic to the index $\mathrm{maj}_R(\sigma)$ in (\ref{Rmaj}) of Reiner, see Lemma \ref{lastlemma}. Under this correspondence, the index $\mathrm{rmaj}(\sigma)$ in (\ref{positive}) is isomorphic to the index $2\,\mathrm{maj}_R(\sigma)-\mathrm{n}_1(\sigma)$ in (\ref{zeng}).
It should be mentioned that we prefer the  R-major index formulation
 based on the the natural order (\ref{norder})
because it is consistent with the order of the flag major index and the negative
major index and it seems to be easier to describe.

In this paper, we extend the three indices on signed permutations, i.e., the inversion number, the flag major index,
and the R-major index,  to signed labeled forests. The inversion number for signed labeled forests is motivated by the inversion number (\ref{BWinv}) and the length function (\ref{LFB}).  The flag major index for
signed labeled forests is an extension of the major indices (\ref{BWmaj}) for forests and (\ref{flagmaj}) for signed permutations, whereas the R-major index for signed labeled forests stems from the major indices  (\ref{BWmaj}) and (\ref{positive}).

 We obtain $q$-hook length formulas by $q$-counting  signed labelings of a given forest with respect to the above indices, from which we see that these three indices are equidistributed for signed labeled forests. Our formulas for the major indices and the inversion number are  type $B$ analogues of the formula due to Bj\"{o}rner and Wachs. We also define the inversion number on even-signed labeled forests in connection with the inversion number (\ref{BWinv}) and the length function (\ref{LFD}) for even-signed permutations.

This paper is organized as follows. In Section 2, we give the definitions of the inversion number and the two major indices on singed labeled forests. The main results are described in this section. We also include a sketch of the proof of the $q$-hook length  formula for the flag major index for signed labeled forests.  Section 3 is   devoted to the
proofs of the generating function formulas for the inversion numbers on signed labeled forests and even-signed labeled forests. In Section 4, we consider the generating function of the R-major index. To this end, we define
$(P,w)$-partitions of type $B$ which can be viewed as a type $B$ extension of the usual
$(P,w)$-partitions as introduced by Stanley \cite{Stanley2}. In Section
5, we give a bijection which establishes the connection  between
the flag  major index and the R-major index on signed labeled forests. Section 6 gives some further questions.

\section{Main results}

In this section, we give the definitions of the inversion number, the flag
 major index and the R-major index on singed (even-signed) labeled forests as aforementioned  in the introduction, and outline the main results of this paper.

Let $F$ be a forest.
Denote by $B_n(F)$ (resp., $D_n(F)$) the set of  signed (resp., even-signed)
labeled forests with the underlying forest $F$. We use $(F,w)$ to
denote a signed labeling of $F$.  The motivation to consider types $B$ and $D$ analogues for signed  labeled
forests  is  the observation that the number $\mathrm{n_2}(\sigma)$ in (\ref{N2}) has a
natural extension to  signed labeled forests; that is, for
$(F,w)\in B_n(F)$, we may define
\begin{equation}\label{Bextension}
\mathrm{n}_2(F,w)=|\{(u,v)\colon u<_Fv, w(u)+w(v)<0\}|.
\end{equation}
Let $\mathrm{n}_1(F,w)$ be the number of negative labels of $w$. So we define the inversion number
for $(F,w)\in B_n(F)$ as
\begin{equation}\label{SLFinv}
\mathrm{inv}_B(F,w)=\mathrm{inv}(F,w)+\mathrm{n}_1(F,w)+\mathrm{n}_2(F,w),
\end{equation}
and  for $(F,w)\in D_n(F)$ the inversion number is defined by
\begin{equation}\label{ESLFinv}
\mathrm{inv}_D(F,w)=\mathrm{inv}(F,w)+\mathrm{n}_2(F,w).
\end{equation}

The first two theorems assert that the inversion numbers (\ref{SLFinv}) and (\ref{ESLFinv}) lead to types $B$ and $D$ analogues of the length generating functions (\ref{lenB}) and (\ref{lenD1}) respectively.

\begin{theorem}\label{ThminvB}
Let $F$ be a forest of size $n$. Then
\begin{equation}\label{lenB2}
\sum_{(F,\,w)\in\, B_n(F)}q^{\mathrm{inv}_B(F,\,w)}=\frac{n!}{\prod_{u\in F}h_u}\prod_{u\in
F}[2h_u].\end{equation}
\end{theorem}

\begin{theorem}\label{ThminvD}
Let $F$ be a forest of size $n$. Then
\begin{equation}\label{sinvD}
\sum_{(F,\,w)\in\, D_n(F)}q^{\mathrm{inv}_D(F,\,w)}=\frac{n!}{2\prod_{u\in
F}h_u}\prod_{u\in F}(1+q^{h_u-1})[h_u].\end{equation}
\end{theorem}

Note that Theorem \ref{LGFB} and Theorem \ref{LGFD} can be deduced respectively from Theorem \ref{ThminvB} and Theorem \ref{ThminvD} by restricting  $F$ to  a linear order.

We next turn to two major  indices for signed labeled forests. The first  is based on the flag major index for signed permutations, namely, for
$(F,w)\in B_n(F)$, we define
\begin{equation}\label{SLFfmaj}
\mathrm{fmaj}(F,w)=2\,\mathrm{maj}(F,w)+\mathrm{n}_1(F,w).
\end{equation}
The following theorem shows that the flag major index (\ref{SLFfmaj}) is equidistributed with the inversion number (\ref{SLFinv}) for signed labeled
forests.

\begin{theorem}\label{Tfmaj}
Let $F$ be a forest of size $n$. Then
\begin{equation}\label{fmaj}
\sum_{(F,\,w)\in\, B_n(F)}q^{\mathrm{fmaj}(F,\,w)}=\frac{n!}{\prod_{u\in
F}h_u}\prod_{u\in F}[2h_u].
\end{equation}
\end{theorem}

On the other hand, it will be seen  that Theorem \ref{Tfmaj} can be deduced from  Theorem \ref{TBJ1} and
a decomposition of the signed permutation group. We sketch the proof below.
Let \[T_n=\{\sigma_1\cdots\sigma_n\in B_n\colon\, \sigma_1<\cdots<\sigma_n\}.\]
Then the signed permutation group $B_n$ has the following decomposition
\begin{equation}\label{decomposition}
B_n=\bigcup_{\pi\in S_n}\{\sigma\pi\colon\, \sigma\in T_n\},
\end{equation}
which is the multiplicative
decomposition of a Coxeter group into a parabolic subgroup and its minimal coset
representatives, see, e.g., Bj\"{o}rner and Brenti \cite{BB} or Humphreys \cite{Humphreys}. Such a decomposition has been used by Adin et al. \cite[Proposition 3.1]{ABR} to prove the Mahonian property of the negative major index (\ref{nmaj}).

\noindent\textit{Sketch of the proof of Theorem \ref{Tfmaj}.} Let $w_0$ be
 a decreasing labeling of $F$, i.e., a labeling of $F$ such that $w_0(u)>w_0(v)$ whenever $u>v$. We now define a bijection  $\psi\colon (F,w)\longmapsto \sigma$  from  $B_n(F)$ to $B_n$ such that $\sigma_i=w(i)$ for $1\leq i\leq n$. Let
\[T_n(F)=\{\psi^{-1}(\sigma)\colon\, \sigma\in T_n\},\]
and let $U_n(F)=\psi^{-1}(S_n)$ be the set of ordinary  labelings of $F$. From the decomposition (\ref{decomposition}), we get
\begin{equation}\label{decompsition2}
B_n(F)=\bigcup_{w\in U_n(F)}\{\tau w\colon\, \tau\in T_n(F)\},
\end{equation}where $\tau w(i)=\tau (w(i))$ for any vertex $i\in F$.

Let $u$ be the parent of $v$, and let  $w_0(u)=i_{u}$ and $w_0(v)=i_v$. It is not hard to verify that $\tau w(i_u)>\tau w(i_v)$ if and only if $w(i_u)>w(i_v)$. This  implies that $\tau w$ and $w$ have the same descent set. Similarly, we see that  $\mathrm{n_1}(F,\,\tau w)=\mathrm{n_1}(F,\,\tau)$. Thus
\begin{equation}\label{key}
\begin{split}
\mathrm{fmaj}(F,\,\tau w)&=2\,\mathrm{maj}(F,\,\tau w)+\mathrm{n_1}(F,\,\tau w)\\
&=2\,\mathrm{maj}(F,\,w)+\mathrm{n_1}(F,\,\tau).
\end{split}
\end{equation}
Again, by the decomposition (\ref{key}) and  Theorem \ref{TBJ1}, we obtain
\begin{align*}
\sum_{(F,\,w)\in B_n(F)}q^{\mathrm{fmaj}(F,\,w)}
&=\sum_{w\in U_n(F)}\sum_{\tau\in T_n(F)}q^{2\,\mathrm{maj}(F,\,w)+\mathrm{n_1}(F,\,\tau)} \\[5pt]
&=\sum_{\tau\in T_n(F)}q^{\mathrm{n_1}(F,\,w)}\ \frac{n!}{\prod_{u\in
F}h_u}\prod_{u\in
F}[h_u]_{q^2} \\[5pt]
&=(1+q)^n\frac{n!}{\prod_{u\in
F}h_u}\prod_{u\in
F}[h_u]_{q^2}\\[5pt]
&=\frac{n!}{\prod_{u\in
F}h_u}\prod_{u\in
F}(\left[h_u]_{q^2}(1+q)\right)\\[5pt]
&=\frac{n!}{\prod_{u\in
F}h_u}\prod_{u\in
F}[2h_u],
\end{align*}
as desired. \qed

The second major index  for  signed labeled forests is the R-major index. For $(F,w)\in B_n(F)$,  we define the
descent set  as
\[\mathrm{Des}_B(F,w)=\mathrm{Des}(F,w)\bigcup\,\{u\in F\colon\, \mbox{$u$ is a root of $F$ with a positive label}\}.\]
Let
\begin{equation}\label{major2}
\mathrm{maj}_B(F,w)=\sum_{u\in \mathrm{Des}_B(F,w)}h_u.
\end{equation}
Denote by $\mathrm{p}(F,w)$ the number of positive labels of $w$. Then the R-major index is defined by
\begin{equation}\label{SLFrmaj}
\mathrm{rmaj}(F,w)=2\,\mathrm{maj}_B(F,w)-\mathrm{p}(F,w).
\end{equation}
The following theorem shows that the R-major index is
equidistributed with the inversion number  for signed labeled forests.

\begin{theorem}\label{Tnmaj}
Let $F$ be a forest of size $n$. Then
\begin{equation}\label{nnmaj}
\sum_{(F,\,w)\in\, B_n(F)}q^{\mathrm{rmaj}(F,\,w)}=\frac{n!}{\prod_{u\in
F}h_u}\prod_{u\in F}[2h_u].
\end{equation}
\end{theorem}

\section{The inversion numbers for types $B$ and $D$}

In this section, we are aimed to prove Theorems \ref{ThminvB} and \ref{ThminvD}, both of which  can be deduced from  the following  theorem involving the weighted counting  of signed labeled forest with respect to the number of inversions and the number of negative labels.

\begin{theorem}\label{thmsinv}
Let $F$ be a forest of size $n$. Then
\begin{equation}\label{sinv}
\sum_{(F,\,w)\in\, B_n(F)}t^{\mathrm{n}_1(F,\,w)}q^{\mathrm{inv}_B(F,\,w)}=\frac{n!}{\prod_{u\in
F}h_u}\prod_{u\in F}(1+tq^{h_u})[h_u].\end{equation}
\end{theorem}

It is clear that  the constant terms on both sides of (\ref{sinv}) lead to
the $q$-hook length formula of Bj\"orner and Wachs as stated in Theorem \ref{TBJ1}.

\pf We proceed by induction on $n$. Assume that  $F$ is a forest consisting of $k$ trees $T_1,\ldots,T_k$. Let us consider the following two cases.

Case 1. Suppose that $k>1$ and  $T_i$ has $n_i$ vertices for  $1\leq i\leq
k$. Each signed labeling of $F$ corresponds  to a
$k$-tuple consisting of a distribution of the $n$ labels (some of
which may have minus signs) into $k$ trees with each tree $T_i$
receiving $n_i$ labels. We see that
\begin{equation*}
\sum_{(F,\,w)}t^{\mathrm{n}_1(F,w)}q^{\mathrm{inv}_B(F,w)}={n\choose
{n_1,\ldots,n_k}}\prod_{i=1}^k
\sum_{(T_i,\,w_i)}t^{\mathrm{n}_1(T_i,w_i)}q^{\mathrm{inv}_B(T_i,w_i)}.
\end{equation*}
Applying the induction hypothesis yields
\begin{align*}
\sum_{(F,\,w)}t^{\mathrm{n}_1(F,w)}q^{\mathrm{inv}_B(F,w)}&={n\choose
{n_1,\ldots,n_k}}\prod_{i=1}^k\frac{n_i!}{\prod_{u\in
T_i}h_u}\prod_{u\in T_i}(1+tq^{h_u})[h_u]\\
&=\frac{n!}{\prod_{u\in F}h_u}\prod_{u\in F}(1+tq^{h_u})[h_u].
\end{align*}

 Case 2. Suppose that $k=1$. Let $F'$ be the
forest obtained from $F$ by removing the root $u$. Every signed
labeling $w$ of $F$ corresponds  to a pair $(w(u),w')$, where $w'$ is the signed labeling of $F'$ induced by $w$. Since $w$ is a signed labeling, we need to
consider the two subcases $w(u)>0$ and $w(u)<0$.

Assume that $w(u)>0$. Then let $w(u)=i$, where $1\leq i\leq n$. We get
\begin{align*}
\mathrm{inv}(F,w)&=\mathrm{inv}(F',w')+|\{v\in F'\colon w'(v)>i\}|,\\[5pt]
\mathrm{n}_1(F,w)&=\mathrm{n}_1(F',w'),\\[5pt]
\mathrm{n}_2(F,w)&=\mathrm{n}_2(F',w')+|\{v\in F'\colon w'(v)+i<0\}|.
\end{align*}
It is easy to check
\[\{v\in F'\colon w'(v)>i\}\cup\{v\in F'\colon w'(v)+i<0\}=\{v\colon i<|w'(v)|\leq n\},\] where $\cup$ denotes the disjoint union. Thus we find
\[\mathrm{inv}_B(F,w)=\mathrm{inv}_B(F',w')+n-i.\]

 Assume that $w(u)<0$. Then let $w(u)=-i$, where $1\leq i\leq n$. We have
\begin{align*}
\mathrm{inv}(F,w)&=\mathrm{inv}(F',w')+|\{v\in F'\colon w'(v)>-i\}|,\\[5pt]
\mathrm{n}_1(F,w)&=\mathrm{n}_1(F',w')+1,\\[5pt]
\mathrm{n}_2(F,w)&=\mathrm{n}_2(F',w')+|\{v\in F'\colon w'(v)-i<0\}|.
\end{align*}
We need to determine the numbers $c_1$ and $c_2$ defined by
\[ c_1=|\{v\in F'\colon w'(v)>-i\}|,\quad   c_2=|\{v\in
F'\colon w'(v)-i<0\}|.\]
\begin{itemize}
\item[(i).] Consider the index  $j$ with $i<j \leq n$. If $j$ is in the labeling set of $w$,  then $j$
contributes $1$ to $c_1$ and $0$ to  $c_2$. If $-j$
is in the labeling set of $w$, then $-j$ contributes $0$ to
$c_1$ and $1$ to  $c_0$.
\item[(ii).]
Consider the index $j$ with $1\leq j<i$. If $j$ is in the labeling set of $w$, then $j$
contributes $1$  to both $c_1$ and $c_2$.  If $-j$ is in the labeling set of $w$, then $-j$ also
contributes $1$ to both $c_1$ and $c_2$.
\end{itemize}
\allowdisplaybreaks
It follows that
\begin{equation*}
|\{v\in F'\colon w'(v)>-i\}+|\{v\in F'\colon
w'(v)-i<0\}|=(n-i)+2(i-1)=n-2+i,
\end{equation*}
which gives  \begin{align*} \mathrm{inv}_B(F,w)&=\mathrm{inv}_B(F',w')+c_1+c_2+1\\[5pt]
&=\mathrm{inv}_B(F',w')+n-1+i.\end{align*} Combining the above two cases $w(u)>0$ and $w(u)<0$ and using the induction hypothesis, we deduce that
\begin{align*}
&\sum_{(F,\,w)}t^{\mathrm{n}_1(F,\,w)}q^{\mathrm{inv}_B(F,\,w)}\\[5pt]
=&\sum_{i=1}^nq^{n-i}\sum_{(F',\,w')}t^{\mathrm{n}_1(F',\,w')}q^{\mathrm{inv}_B(F',\,w')}
+\sum_{i=1}^ntq^{n-1+i}\sum_{(F',\,w')}t^{\mathrm{n}_1(F',\,w')}q^{\mathrm{inv}_B(F',\,w')}\\[5pt]
=&(1+q+\cdots+q^{n-1}+t(q^n+\cdots+q^{2n-1}))\sum_{(F',\,w')}t^{\mathrm{n}_1(F',\,w')}q^{\mathrm{inv}_B(F',\,w')},\\[5pt]
=&(1+tq^n)[n]\sum_{(F',\,w')}t^{\mathrm{n}_1(F',\,w')}q^{\mathrm{inv}_B(F',\,w')}\\[5pt]
=&(1+tq^n)[n]\frac{(n-1)!}{\prod_{u\in F'}h_u}\prod_{u\in
F'}(1+tq^{h_u})[h_u]\\[5pt]
=&\frac{n!}{\prod_{u\in F}h_u}\prod_{u\in F}(1+tq^{h_u})[h_u],
\end{align*}
as claimed.  \qed

Once Theorem \ref{thmsinv} is established, it is easy to derive Theorem \ref{ThminvB} by taking $t=1$ in (\ref{sinv})
and by observing that
\[(1+q^n)[n]=[2n].\]

We are now ready to give a proof of Theorem \ref{ThminvD}.

\noindent\textit{Proof of Theorem \ref{ThminvD}.}  Let
\begin{equation*}
D_n(t,q)=\sum_{(F,\,w)\in B_n(F)}t^{\mathrm{n}_1(F,\,w)}q^{\mathrm{inv}(F,w)+\mathrm{n}_2(F,w)}.
\end{equation*}
By Theorem \ref{thmsinv} we obtain
\begin{align*}
D_n(t,q)=&\sum_{(F,\,w)\in B_n(F)}(t/q)^{\mathrm{n}_1(F,\,w)}q^{\mathrm{inv}(F,w)+\mathrm{n}_1(F,w)+\mathrm{n}_2(F,w)}\\[5pt]
=&\sum_{(F,\,w)\in B_n(F)}(t/q)^{\mathrm{n}_1(F,\,w)}q^{\mathrm{inv}_B(F,\,w)}\\[5pt]
=&\frac{n!}{\prod_{u\in F}h_u}\prod_{u\in F}(1+tq^{h_u-1})[h_u].
\end{align*}
Since $F$ has at least one leaf, there must exist a factor $(1+tq^{h_u-1})=(1+t)$  in $D_n(t,q)$. Hence $D_n(-1,q)=0$, or, equivalently,
\begin{equation}\label{even}
\sum_{\mbox{$i$ is even}}[t^i]D_n(t,q)=\sum_{\mbox{$i$ is odd}}[t^i]D_n(t,q),\end{equation}where $[t^i]$ denotes the coefficient of $t^i$ in $D_n(t,q)$.
Thus
\begin{align*}
\sum_{(F,\,w)\in\, D_n(F)}q^{\mathrm{inv}_D(F,\,w)}&=\sum_{\mbox{$i$ is even}}[t^i]D_n(t,q)\\[5pt]
&=\frac{D_n(1,q)}{2},\end{align*} which coincides with the formula (\ref{sinvD}). \qed

It would be interesting to give a combinatorial interpretation for (\ref{even}).

\section{The R-major index for signed labeled forests}

In this section, we shall present a proof of Theorem \ref{Tnmaj}. This theorem will be deduced from the following more general formula.

\begin{theorem}\label{Thm1}
Let $F$ be a forest of size  $n$. Then
\begin{equation}\label{main}
\sum_{(F,w)\in\, B_n(F)}t^{\mathrm{p}(F,w)}q^{\mathrm{maj}_B(F,w)}=\frac{n!}{\prod_{u\in
F}h_u}(1+tq)^n \prod_{u\in F}[h_u].\end{equation}
\end{theorem}

 In fact, the proof of the above Theorem relies on
 the formula for $q$-counting the major index $\mathrm{maj}_B$ of  linear extensions of a signed labeled forest, which can be considered as type $B$
  analogue of the $q$-hook length formula  of Bj\"{o}rner and Wachs \cite{BW} for the $q$-counting of the major index of linear extensions.

Let us recall the definition of linear extensions of poset. For a poset $P$ with $n$ elements $x_1,\ldots,x_n$, linear extensions of $P$ can be seen as  permutations $x_{i_1}\cdots x_{i_n}$ such that   $x_{i_j}<_Px_{i_k}$ implies $j<k$.  A labeling of $P$ is a bijection from $\{x_1,\ldots, x_n\}$ to $[1,n]$. For a poset  $P$ with a labeling $w$, we usually use a permutation $w(x_{i_1})\cdots w(x_{i_n})$  to represent  linear extension $x_{i_1}\cdots x_{i_n}$. Denote by $\mathcal{L}(P,w)$ the set of all such permutations. Bj\"{o}rner and Wachs \cite{BW} obtained  the following generating function for the major index of linear extensions of any given labeled forest, which reduces to the result of Stanley \cite{Stanley2} when the labeling is decreasing.

\begin{theorem}[\mdseries{Bj\"{o}rner and Wachs \cite{BW}, Theorem
1.2}]\label{Thm3}Let $F$ be a forest of size $n$, and $w$ a labeling
of $F$. Then
\begin{equation}\sum_{\pi \in
\mathcal{L}(F,w)}q^{\mathrm{maj}(\pi)}=q^{\mathrm{maj}(F,w)}\frac{[n]!}{\prod_{u_\in
F}[h_u]}. \end{equation}
\end{theorem}

When $w$ is a signed labeling,  $\mathcal{L}(F,w)$, as defined above, is a set of signed permutations.
We obtain the following theorem which can be viewed as a type $B$ analogue of Theorem \ref{Thm3}.

\begin{theorem}\label{Le1}
Let $F$ be a forest of size $n$, and $w$ a signed labeling of $F$. Then
\begin{equation}\label{Le1equ}
\sum_{\sigma \in
\mathcal{L}(F,w)}q^{\mathrm{maj}_B(\sigma)}=q^{\mathrm{maj}_B(F,w)}\frac{[n]!}{\prod_{u\in
F}[h_u]}.\end{equation}
\end{theorem}

Though Theorem \ref{Le1} is a type $B$ analogue of Theorem \ref{Thm3}, its proof does not seem to be straightforward.  To prove Theorem \ref{Thm3}, Bj\"{o}rner and Wachs \cite{BW} defined the recursive labelings for forests. They first derived  the following $q$-hook length formula
\[\sum_{\pi \in
\mathcal{L}(F,w)}q^{\mathrm{inv}(\pi)}=q^{\mathrm{inv}(F,w)}\frac{[n]!}{\prod_{u_\in
F}[h_u]},\]
where $w$ is a recursive labeling, see Bj\"{o}rner and Wachs \cite[Theorem 1.1]{BW}. Then they proved that Foata's bijection (see, Foata \cite{Foata}) is invariant when restricted to the set of linear extensions of a forest with a recursive labeling, see Bj\"{o}rner and Wachs \cite[Theorem 2.2]{BW}. Moreover they observed that the inversion number and the major index of a labeled forest are equal for a recursive labeling, see Bj\"{o}rner and Wachs \cite[Lemma 2.3]{BW}. For recursive labelings,  they obtained Theorem \ref{Thm3}. Finally,
 they used a equivalence relation on labelings to extend the special case to the general case, see Bj\"{o}rner and Wachs \cite[Lemma 4.2]{BW}, and eventually  finished the proof of Theorem \ref{Thm3}.

The way in which Bj\"{o}rner and Wachs \cite{BW} proved Theorem \ref{Thm3} does not seem to apply to Theorem \ref{Le1}.  To prove Theorem \ref{Le1}, we introduce  $(F,w)$-partitions of type $B$ which reduce to the $(P,w)$-partitions due to Stanley \cite{Stanley2} when $P$ is a forest  and $w$  is a signed  labeling of $P$ with labeling set $\{-1,-2,\ldots,-n\}$.  When $w$ is an ordinary labeling,  our approach leads to a direct combinatorial  proof of Theorem \ref{Thm3} of Bj\"{o}rner and Wachs.

Let $\mathbb{N}$ be the set of nonnegative integers. A  $(F,w)$-partition of type $B$ is a map $f\colon V(F) \longrightarrow
\mathbb{N}$ satisfying the following conditions

\begin{itemize}
\item[(1)] $f(x)\leq f(y)$ if $x\geq_F y$;
\item[(2)] $f(x)<f(y)$ if $x>_Fy$ and $w(x)<w(y)$;
\item[(3)] $f(u)\geq 1$ if $u$ is a root of $F$ with $w(u)>0$.
\end{itemize}
We denote by $A_B(F,w)$ the set of  $(F,w)$-partitions of type $B$. For $f\in A_B(F,w)$, let
\[ |f|=\sum_{x\in F}f(x).\]

We shall compute the generating function for the $q$-counting of
$(F,w)$-partitions of  type $B$ in two different ways.  These two different countings lead to the following identity
\begin{equation}\label{relation}
\frac{\sum_{\sigma\in\mathcal{L}(F,w)}q^{\mathrm{maj}_B(\sigma)}}{(1-q)(1-q^2)\cdots
(1-q^n)}=\frac{q^{\mathrm{maj}_B(F,w)}}{\prod_{u\in
F}(1-q^{h_u})},\end{equation} which is equivalent to the formula (\ref{Le1equ}).

\begin{lemma}\label{Lem3}
Let $F$ be a forest and $w$ a signed labeling of $F$. Then
\begin{equation}\label{hook}
\sum_{f\in A_B(F,w)}q^{|f|}=\frac{q^{\mathrm{maj}_B(F,w)}}{\prod_{u\in
F}(1-q^{h_u})}.\end{equation}
\end{lemma}

\pf For each $f\in A_B(F,w)$ and $u\in \mathrm{Des}_B(F,w)$, define $f_u\colon\ V(F) \longrightarrow
\mathbb{N}$ as
\begin{equation*}
f_u(x)=\left\{\begin{array}{ll}
f(x)-1, &\mbox{if $x\leq_F u$},\\[3pt]
f(x),&\mbox{otherwise}.\end{array}\right.
\end{equation*}
Assume $\mathrm{Des}_B(F,w)=\{u_1,u_2,\ldots,u_k\}$. Define recursively $f_{u_1u_2\cdots u_k}=(f_{u_1u_2\cdots u_{k-1}})_{u_k}$. Since $f_{uv}=f_{vu}$ then $f_{u_1u_2\cdots u_k}$ is independent of the order of $u_1,\ldots,u_k$. Thus $f_{u_1u_2\cdots u_k}$ is well defined. It is easy to see the resulting $f_{u_1u_2\cdots u_k}$ are maps from $V(F)$ to $\mathbb{N}$ satisfying only condition (1). Such maps are formally called  $P$-partitions where $P$ is a forest, see Stanley \cite[Chapter 4]{Stanley1}. In such a way,  for fixed $(F,w)$ we
establish a bijection  $\varphi\colon f\longmapsto f_{u_1u_2\cdots u_k}$ between the
following two sets\begin{equation}\label{onetoone} \varphi\colon\
A_B(F,w)\longrightarrow \{\mbox{$F$-partitions}\},\end{equation}which,
for each $f\in A_B(F,w)$, satisfies
\begin{align}|f|&=\sum_{u\in
\mathrm{Des}_B(F,\,w)}h_u+|\varphi(f)|\nonumber\\[5pt]
&=\mathrm{maj}_B(F,w)+|\varphi(f)|.\label{rela1}
\end{align}
Stanley \cite[Proposition 22.1]{Stanley2} has proved (see Sagan \cite{Sagan} for a combinatorial proof)
\begin{equation}\label{gene1}\sum_{g}q^{|g|}=\frac{1}{\prod_{u\in F}(1-q^{h_u})},\end{equation} where
the sum ranges over all $F$-partitions.
(\ref{rela1})  together with  (\ref{gene1})  implies (\ref{hook}). This completes the proof. \qed

Lemma \ref{Lem3} gives one formulation for the generating function of $A_B(F,w)$. To give the other, we go on by extending  a fundamental result concerning the usual $(P,w)$-partitions. For $\sigma=\sigma_1\cdots\sigma_n\in B_n$, a map
$f\colon \{\sigma_1,\ldots,\sigma_n\}\longrightarrow \mathbb{N}$ is said to be $\sigma$-compatible if it satisfies the following conditions
\begin{itemize}
\item[(i)] $f(\sigma_1)\geq f(\sigma_2)\geq\cdots\geq f(\sigma_n)$;
\item[(ii)] For $i\in [1,n-1]$, $f(\sigma_i)>f(\sigma_{i+1})$ if $\sigma_i>\sigma_{i+1}$;
\item[(iii)] $f(\sigma_n)\geq 1$ if $\sigma_n>0$.
\end{itemize}
Let $A_{\sigma}^B$ denote the set of all $\sigma$-compatible maps.  Then we have the following decomposition.

\begin{lemma}\label{decompsition3}
Let $F$ be a forest with a signed labeling $w$. Then
\begin{equation}\label{dec1}
A_B(F,w)=\biguplus_{\sigma\in
\mathcal{L}(F,w)}A_\sigma^B.\end{equation}
\end{lemma}

Note that if $w$ is  a signed  labeling with the labeling set $\{-1,-2,\ldots,-n\}$ then Lemma \ref{decompsition3} coincides with
 the decomposition of Stanley \cite[Theorem 6.2]{Stanley2} where the poset is a forest.

\pf  By the definition of $(F,w)$-partitions of type $B$ and the decomposition of Stanley \cite[Theorem 6.2]{Stanley2}, it is not hard to see  for each $f\in A_B(F,w)$ there exists a unique
linear extension $\sigma\in \mathcal{L}(F,w)$ such that $f\in A_\sigma^B$.
It suffices to verify that for each  $\sigma\in \mathcal{L}(F,w)$, if   $f\in A_\sigma^B$ then $f\in A_B(F,w)$. In other words, it is necessary to show that $f(\sigma_i)\geq 1$ whenever $\sigma_i>0$. Note that condition (iii) ensures this holds  in the case of $\sigma_n>0$. So we are left with  the case  $\sigma_n<0$. Assuming $\sigma_n<0$, let $\sigma_j$ be the rightmost entry of $\sigma$ such that $\sigma_j>0$. Then we have $\sigma_j>\sigma_{j+1}$ since $\sigma_{j+1}<0$. Given condition (ii), we see that $f(\sigma_j)\geq 1$. Therefore we have  $f(\sigma_i)\geq 1$ whenever $\sigma_i>0$. This completes the proof. \qed

\begin{lemma}\label{Lem2}
Let $F$ be a forest and $w$ a  signed labeling of $F$. Then
\begin{equation}\label{abf}
\sum_{f\in A_B(F,w)}q^{|f|}=\frac{\sum_{\sigma\in\mathcal{L}(F,w)}q^{\mathrm{maj}_B(\sigma)}}{(1-q)(1-q^2)\cdots (1-q^n)}.
\end{equation}
\end{lemma}

\pf For $f\in A_\sigma^B$ and
 $1\leq i\leq n-1$, let
\[ p_i=f(\sigma_i)-f(\sigma_{i+1}),\]
 and let $p_n=f(\sigma_n)$. Clearly,
\[|f|=p_1+2p_2+\cdots+np_n.\]
Under the conditions (i), (ii), and (iii), we see that $p_i>0$ if $i\in
\mathrm{Des}_B(\sigma)$ and $p_i\geq 0$ otherwise. Hence
\begin{align*}
\sum_{f\in
A_\sigma^B}q^{|f|}&=\sum_{(p_1,\ldots,p_n)}q^{p_1+2p_2+\cdots+np_n}=q^{\mathrm{maj}_B(\sigma)}\sum_{(q_1,\ldots,q_n)\atop
{q_i\geq
0}}q^{q_1+2q_2+\cdots+nq_n}\\
&=\frac{q^{\mathrm{maj}_B(\sigma)}}{(1-q)(1-q^2)\cdots(1-q^n)}.
\end{align*}
Now, using the decomposition  (\ref{dec1}), we get (\ref{abf}). \qed

Combining
Lemma
\ref{Lem3} and  Lemma \ref{Lem2}, we deduce  Theorem \ref{Le1}.
We now need one more lemma for the proof of Theorem \ref{Thm1}, which shows that the R-major index for singed permutations is equidistributed with Reiner's major index  (\ref{Rmaj}).

\begin{lemma}\label{lastlemma} For $n\geq 0$, we have
\begin{equation} \label{sbn}
\sum_{\sigma\in B_n}t^{\mathrm{p}(\sigma)}q^{\mathrm{maj}_B(\sigma)}=(1+tq)^n[n]!.
\end{equation}
\end{lemma}
\pf  We will give a bijection $\psi\colon \sigma\longmapsto \tau$ on $B_n$  such that $\mathrm{maj}_B(\sigma)=\mathrm{maj}_R(\tau)$ and $\mathrm{p}(\sigma)=\mathrm{n}_1(\tau)$. Then the formula (\ref{sbn})
 follows from Reiner's formula (\ref{rg}).  For $\sigma=\sigma_1\cdots \sigma_n\in B_n$, let $\{\sigma_{i_1},\ldots,\sigma_{i_k}\}_<$ (resp., $\{\sigma_{j_1},\ldots,\sigma_{j_{l}}\}_<$) be the set of positive (resp., negative) entries of $\sigma$, where the subscript $_<$ means $\sigma_{i_1}<\cdots<\sigma_{i_k}$. Define $\tau_{i_s}=-\sigma_{i_{k+1-s}}$ for $1\leq s\leq k$, and
$\tau_{j_t}=-\sigma_{j_{l+1-t}}$ for $1\leq t\leq l$. Let $\tau=\tau_1\cdots\tau_n$. It is clear that $\mathrm{p}(\sigma)=\mathrm{n}_1(\tau)$. Then it can be checked that  $\mathrm{Des}_B(\sigma)=\mathrm{Des}_R(\tau)$. The details are omitted.
 Thus $\psi$ is  the required bijection. This completes the proof.
\qed

We are now ready to prove Theorem \ref{Thm1}. We use a similar technique as given by Bj\"{o}rner and Wachs \cite{BW} for deriving the generating function for the major index of a given forest with the ordinary labelings.

\noindent\textit{Proof of Theorem \ref{Thm1}.} We aim to
establish the relation (\ref{main}) by evaluating the double sum
\[\sum_{(F,w)\in B_n(F)}\sum_{\sigma\in \mathcal{L}(F,w)}t^{\mathrm{p}(F,w)}q^{\mathrm{maj}_B(\sigma)}\] in two
different ways.
By Lemma \ref{Le1}, we have
\begin{align*}
\sum_{(F,w)\in B_n(F)}\sum_{\sigma\in
\mathcal{L}(F,w)}t^{\mathrm{p}(F,w)}q^{\mathrm{maj}_B(\sigma)}&=\sum_{(F,w)\in B_n(F)}t^{\mathrm{p}(F,w)}q^{\mathrm{maj}_B(F,w)}\frac{[n]!}{\prod_{u\in
P}[h_u]}\\&=\frac{[n]!}{\prod_{u\in
P}[h_u]}\sum_{(F,w)\in B_n(F)}t^{\mathrm{p}(F,w)}q^{\mathrm{maj}_B(F,w)}.
\end{align*}

On the other hand, we may compute the above double sum by exchanging the order of summation. Let $\chi$ denote the indicator function which equals $1$
when the statement is true and $0$ otherwise. Then we have
\begin{align*}
\sum_{(F,w)\in B_n(F)}\sum_{\sigma\in
\mathcal{L}(F,w)}t^{\mathrm{p}(F,w)}q^{\mathrm{maj}_B(\sigma)}&=
\sum_{(F,w)\in B_n(F)}\sum_{\sigma\in B_n}t^{\mathrm{p}(F,w)}q^{\mathrm{maj}_B(\sigma)}\chi(\sigma\in \mathcal{L}(F,w))\\
&=\sum_{\sigma\in
B_n}q^{\mathrm{maj}_B(\sigma)}\sum_{(F,w)\in B_n(F)}t^{\mathrm{p}(F,w)}\chi(\sigma\in
\mathcal{L}(F,w))\\
&=\sum_{\sigma\in
B_n}q^{\mathrm{maj}_B(\sigma)}t^{\mathrm{p}(\sigma)}\sum_{(F,w)\in B_n(F)}\chi(\sigma\in
\mathcal{L}(F,w)).
\end{align*}
Recall that for any permutation $\pi\in S_n$, \mdseries{Bj\"{o}rner and Wachs
\cite{BW} have shown that there are \[ \frac{n!}{\prod_{u\in F}h_u}\]
ordinary labelings $w$ such that $\pi\in \mathcal{L}(F,w)$. Clearly,
 this counting argument also applies to a signed permutation  $\sigma\in B_n$. Consequently,
\begin{align*}
\sum_{(F,w)\in B_n(F)}\sum_{\sigma\in\mathcal{L}(F,w)}t^{\mathrm{p}(F,w)}q^{\mathrm{maj}_B(\sigma)}&=\sum_{\sigma\in
B_n}q^{\mathrm{maj}_B(\sigma)}t^{\mathrm{p}(\sigma)}\frac{n!}{\prod_{u\in F}h_u}\\
&=\frac{n!}{\prod_{u\in F}h_u}\sum_{\sigma\in
B_n}q^{\mathrm{maj}_B(\sigma)}t^{\mathrm{p}(\sigma)}.
\end{align*}
By Lemma \ref{lastlemma}, we get
\[\sum_{(F,w)\in B_n(F)}\sum_{\sigma\in\mathcal{L}(F,w)}t^{\mathrm{p}(F,w)}q^{\mathrm{maj}_B(\sigma)}
=\frac{n!}{\prod_{u\in F}h_u}(1+tq)^n[n]!.\]

The above double counting gives the following relation
\[\frac{[n]!}{\prod_{u\in
F}[h_u]}\sum_{(F,w)\in B_n(F)}t^{\mathrm{p}(F,w)}q^{\mathrm{maj}_B(F,w)}=\frac{n!}{\prod_{u\in
F}h_u}(1+tq)^n[n]!,\]which is equivalent to (\ref{main}). This completes the proof.
\qed

Based on Theorem \ref{Thm1}, it is easy to derive Theorem \ref{Tnmaj}.

\noindent\textit{Proof of Theorem \ref{Tnmaj}.} Setting $q\rightarrow q^2$ and $t\rightarrow q^{-1}$ in  (\ref{main}), the left-hand side  becomes
\begin{align*}
\sum_{(F,w)\in\, B_n(F)}(q^{-1})^{\mathrm{p}(F,w)}q^{2\mathrm{maj}_B(F,w)}
&=\sum_{(F,w)\in\, B_n(F)}q^{2\mathrm{maj}_B(F,w)-\mathrm{p}(F,w)}\\[5pt]
&=\sum_{(F,w)\in\, B_n(F)}q^{\mathrm{rmaj}(F,w)}
\end{align*} and the right-hand side can be written as
\begin{align*}
\frac{n!}{\prod_{u\in
F}h_u}(1+q^{-1}q^2)^n \prod_{u\in F}[h_u]_{q^2}=\frac{n!}{\prod_{u\in
F}h_u}\prod_{u\in F}[2h_u].\end{align*}
This completes the proof.\qed

\section{A correspondence}

From Theorem \ref{Tnmaj} and Theorem \ref{Tfmaj}, one sees that the R-major index and
the flag major index are equidistributed for signed labeled forests.
One is naturally led to the question of finding a correspondence that
explains the equidistribution property. This is the objective of this section
to provide such a correspondence. Of course, this bijection can be considered
as an alternative proof of Theorem \ref{Tnmaj}.

 Define  a bijection $\phi\colon\ (F,w)\longmapsto
(F,w')$ on $B_n(F)$ as follows. For each vertex $u\in F$,
\begin{itemize}
\item[(1)] $w'(u)$ has the same sign with $-w(u)$,
\item [(2)] $|w'(u)|=n+1-|w(u)|$.
\end{itemize}

\begin{theorem}
The above map $\phi$ is a bijection with the following property \[\mathrm{rmaj}(F,w)=\mathrm{fmaj}(F,w').\]
\end{theorem}

\pf The theorem  holds if we can show that
\begin{equation}\label{relaf-p}
\mathrm{maj}_B(F,w)=\mathrm{maj}(F,w')+\mathrm{p}(F,w)\ \ \mathrm{and}\ \ \mathrm{p}(F,w)=\mathrm{n}(F,w').\end{equation}
We proceed to prove (\ref{relaf-p}) by induction on the number of vertices of $F$.
Without loss of generality, we may assume that  $F$ is a tree.

If $F$ has only one vertex,  it is easy to check (\ref{relaf-p}). So we
 may assume that $F$ has at least two vertices. Let $u_0$ be the root of $F$, and $C(u_0)$ the set of children of $u_0$. By the definition of $\phi$, we see that
  $\mathrm{p}(F,w)=\mathrm{n}(F,w')$. Now we claim that
\begin{equation}\label{ggg}
\mathrm{maj}_B(F,w)=\mathrm{maj}(F,w')+\mathrm{p}(F,w).
\end{equation}
Here are two cases.

Case 1: $w(u_0)>0$.  We partition $C(u_0)$ into the following three subsets.
\begin{align*}
P_>&=\{u\in C(u_0)\colon\, w(u)>w(u_0)\},\\[5pt]
P_<&=\{u\in C(u_0)\colon\, 0<w(u)<w(u_0)\},\\[5pt]
N&=\{u\in C(u_0)\colon\, w(u)<0\}.
\end{align*}
For any vertex $u\in F$, let $F_u$ be the subtree of $F$ rooted at $u$. Then
\begin{equation*}
\begin{split}
&\mathrm{Des}_B(F,w)\\[5pt]
=&\left(\bigcup_{u\in P_>}\mathrm{Des}_B(F_u,w_u)\right)\bigcup\left(\bigcup_{u\in P_<}\mathrm{Des}_B(F_u,w_u)\backslash P_<\right)\bigcup\left(\bigcup_{u\in N}\mathrm{Des}_B(F_u,w_u)\right)\bigcup\,\{u_0\},
\end{split}
\end{equation*}
where $w_u$ is the signed labeling of $F_u$ induced by $w$. So we have
\begin{equation}\label{majBB}
\begin{split}
\mathrm{maj}_B(F,w)=&\sum_{u\in P_>}\mathrm{maj}_B(F_u,w_u)+\sum_{u\in P_<}\mathrm{maj}_B(F_u,w_u)\\[5pt]
&+\sum_{u\in N}\mathrm{maj}_B(F_u,w_u)+n-\sum_{u\in P_<}h_u.
\end{split}
\end{equation}

Let us compute the major index of $w'$.  Let
\begin{align*}
N_>&=\{u\in C(u_0)\colon\, 0>w'(u)>w'(u_0)\},\\[5pt]
N_<&=\{u\in C(u_0)\colon\, w'(u)<w'(u_0)\},\\[5pt]
P&=\{u\in C(u_0)\colon\, w'(u)>0\}.
\end{align*}
Then
\begin{equation*}
\begin{split}
&\mathrm{Des}(F,w')\\[5pt]
=&\left(\bigcup_{u\in N_>}\mathrm{Des}_B(F_u,w'_u)\bigcup N_>\right)\bigcup\left(\bigcup_{u\in N_<}\mathrm{Des}(F_u,w'_u)\right)\bigcup\left(\bigcup_{u\in P}\mathrm{Des}(F_u,w'_u)\bigcup P\right),
\end{split}
\end{equation*}
from which we deduce that
\begin{equation}\label{gg}
\begin{split}
\mathrm{maj}(F,w')=&\sum_{u\in N_>}\mathrm{maj}(F_u,w')+\sum_{u\in N_<}\mathrm{maj}(F_u,w')\\[5pt]
&+\sum_{u\in P}\mathrm{maj}(F_u,w')+\sum_{u\in N_>}h_u+\sum_{u\in P}h_u.
\end{split}
\end{equation}
By the definition of $\phi$, it is not hard to verify
\begin{equation}\label{setset}
N_>=P_>,\ \ N_<=P_<\ \ \mathrm{and}\ \ P=N.
\end{equation}
Therefore, by  (\ref{majBB}), (\ref{gg}), (\ref{setset}), and the induction hypothesis, we find that
\allowdisplaybreaks
\begin{align*}
\mathrm{maj}_B(F,w)=&\sum_{u\in P_>}\left(\mathrm{maj}(F_u,w'_u)+\mathrm{p}(F_u,w_u)\right)+\sum_{u\in P_<}\left(\mathrm{maj}(F_u,w'_u)+\mathrm{p}(F_u,w_u)\right)\\[5pt]
&+\sum_{u\in N}\left(\mathrm{maj}(F_u,w'_u)+\mathrm{p}(F_u,w_u)\right)+n-\sum_{u\in P_<}h_u\\[5pt]
=&\sum_{u\in N_>}\mathrm{maj}(F_u,w'_u)+\sum_{u\in N_<}\mathrm{maj}(F_u,w'_u)
+\sum_{u\in P}\mathrm{maj}(F_u,w'_u)\\[5pt]
&+\mathrm{p}(F,w)-1+n-\sum_{u\in N_<}h_u.\\[5pt]
=&\mathrm{maj}(F,w')+\mathrm{p}(F,w)-1+n-\sum_{u\in N_<}h_u-\sum_{u\in N_>}h_u-\sum_{u\in P}h_u\\[5pt]
=&\mathrm{maj}(F,w')+\mathrm{p}(F,w)-1+n-\sum_{u\in C(u_0)}h_u,
\end{align*}
which reduces to (\ref{ggg}) since
\[\sum_{u\in C(u_0)}h_u=n-1.\]

Case 2:   $w(u_0)<0$. We can use a similar argument to that for Case 1.
So we reach the conclusion that  (\ref{relaf-p}) holds for any tree. This completes the proof.\qed

\section{Concluding remarks}

We conclude this paper with two questions.
 While we have derived the generating functions
 for the flag major index and the R-major index of signed labeled
forests, it would be interesting to give a suitable definition
  of the negative major index (\ref{nmaj}) for signed labeled forests.
Intuitively,  a  natural choice would be
\begin{equation}\label{snmaj1}
\mathrm{nmaj}(F,w)=\mathrm{maj}(F,w)+\mathrm{n}_1(F,w)+\mathrm{n}_2(F,w).
\end{equation}
However, the above statistic is not  equidistributed with  the inversion number (\ref{SLFinv}).

For type $D_n$ permutations, Biagioli \cite{Biagioli2} defined the  major index as
follows
\begin{equation}\label{dmaj}
\mathrm{dmaj}(\sigma)=\mathrm{maj}(\sigma)+\mathrm{n}_2(\sigma),
\end{equation} and shown it is equidistributed with the length function of $D_n$. However, the following statistic
\begin{equation}\label{snmaj2}
\mathrm{dmaj}(F,w)=\mathrm{maj}(F,w)+\mathrm{n}_2(F,w)
\end{equation}
for labeled forests of type $D_n$   is not equidistributed with
the inversion number (\ref{ESLFinv}) of $D_n(F)$.
 We would pose the question of finding an appropriate major index
 for $D_n(F)$ which is equidistributed with the inversion number.

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Copyright © 2018 Xiaofan Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Background. There is a worldwide epidemic of nonorganic gastrointestinal disorders (NOGDs), which are a class of disorders that cause various discomforts and ultimately progress into organic gastrointestinal diseases. Because of the unsatisfactory efficacy of Western medical treatments, traditional Chinese medicine (TCM) is becoming a promising complementary and alternative treatment to manage NOGDs. Objectives. To investigate the efficacy and safety of Hou Gu Mi Xi (HGMX), a newly developed dietary TCM formula, on the syndrome of spleen qi deficiency (SQD) in patients with NOGDs. Methods/Design. This study is a multicenter, randomized, double-blinded, parallel, and placebo-controlled trial that will last for 2 years. All qualified subjects with NOGDs and SQD will be included. The study population will be divided into the HGMX and placebo groups. To assess the efficacy of HGMX, we will mainly focus on changes in SQD symptoms scored by a Spleen Qi Deficiency Symptoms Grading and Quantifying Scale and evaluate changes in gastrin-17, the negative Helicobacter pylori conversion rate, body weight, body mass index, and gastroscopy findings. The safety of HGMX will be assessed by recording adverse events (AEs), severe AEs, treatment-related AEs and withdrawal due to AEs. Discussion. This trial is part of our study series that intends to validate the potential of HGMX in the management of chronic gastrointestinal diseases. This series of RCTs is the first committed to the evaluation of a dietary TCM formula and will hopefully establish an evidence-based clinical research model for dietary TCM formulas. Ethics. The protocol was approved by Ethics Committee of five research hospitals and was registered in Clinicaltrials.gov (NCT03019042).
Chronic gastrointestinal diseases are one of the major health problems worldwide, with an incidence of approximately 60 to 70 million cases per year in the United States . It was reported that up to 38.6 million clinic visits and 45.3 deaths per 100 thousand patients per year are due to chronic gastrointestinal diseases [1, 2]. Meanwhile, the incidence of chronic gastrointestinal diseases in China is 7.3, ranking fifth among all diseases . Pathologically, chronic gastrointestinal diseases can be divided into organic gastrointestinal diseases and nonorganic gastrointestinal disorders (NOGDs); organic gastrointestinal diseases can develop from NOGDs, and a typical routine of pathological progression is from NOGDs to gastrointestinal metaplasia, atypical hyperplasia, precancerous lesions, and, ultimately, cancer [4, 5]. Therefore, although explicit epidemiological data for NOGDs remain unclear, the harm and burden of NOGDs are plausibly substantial.
NOGDs represent a series of disorders, mainly including chronic nonatrophic gastritis and functional gastrointestinal disorders listed in the Rome IV diagnostic criteria . The symptoms of NOGDs are nontypical and may include gastrointestinal discomfort, such as nausea, stomach and abdominal distension, vomiting, acid reflux, diarrhea, and constipation, as well as systemic symptoms [6, 7]. Because it is challenging to cure most organic gastrointestinal diseases, such as atrophic gastritis, peptic ulcers, and gastric cancer, people are focusing on treating NOGDs and preventing progression, which would be more cost-effective than treating organic gastrointestinal diseases [8, 9]. However, the efficacy of most treatment options, predominantly anti-infection, anti-inflammation, and symptomatic treatments, in current Western medicine guidelines for NOGDs is unsatisfactory [10, 11].
Along with the development of medical science, traditional Chinese medicine (TCM) is playing an increasingly important role in the treatment of chronic gastrointestinal diseases, especially NOGDs . The common symptoms of NOGDs mentioned above can be categorized as six syndromes in TCM—spleen qi deficiency (SQD), spleen-stomach deficiency cold, spleen deficiency and dampness, spleen deficiency and liver depression, liver and spleen imbalance, and liver qi stagnation—among which SQD is the most common, and the primary symptoms include stomach and abdominal distension, fatigue, weakness, loss of appetite, and diarrhea . Spleen and qi are concepts in TCM that play roles in transiting food and water and maintaining activity, respectively. The TCM-based pathogenesis of spleen qi-deficient NOGDs is dysfunction in the spleen and stagnation of qi, which cause blood stasis, accumulation of phlegm, and food retention. Therefore, TCM therapy for NOGDs aims to invigorate and replenish qi, supported by nourishing the spleen, activating blood circulation, and removing damp evil, qi stagnation, and food retention .
Shen Ling Bai Zhu San, a classic Chinese herbal medicine formula, was originally described in Tai Ping Hui Min He Ji Ju Fang in the Nang Song Dynasty (1102 AD) . Because of its effects of replenishing qi and nourishing the spleen, it has been used to treat SQD for thousands of years, especially for patients with NOGDs . However, Shen Ling Bai Zhu San is a poor tasting powder that must be mixed with boiled water, which restricts its widespread use and long-term compliance.
On the basis of Shen Ling Bai Zhu San, a dietary TCM formula named Hou Gu Mi Xi (HGMX) is under development by Jiangzhong Pharmaceutical (group) Co., Ltd., Jiangxi, China, and had come into the market in September 2016, in which atractylodes (an herb that is forbidden in food) is replaced with orange peel. Consequently, HGMX is composed of ginseng (Renshen), tuckahoe (Fuling), coixenolide (Yiyiren), Chinese yam (Shanyao), lotus seed (Lianzi), amomum (Sharen), platycodon (Jiegen), white hyacinth bean (Baibiandou), licorice (Gancao), and orange peel (Jupi), all in the list of food in China. In this formula, ginseng, tuckahoe, coixenolide, and Chinese yam can invigorate spleen qi, remove dampness, and nourish our body, with the support of lotus seed, amomum, platycodon, white hyacinth bean, licorice, and orange peel. Theoretically, HGMX can replenish the spleen and stomach qi, remove the dampness and turbidity, and dissipate the qi stagnation such that the spleen qi can run normally and the damp evil can be removed, leading to the natural elimination of the symptoms of SQD. In addition, HGMX has been made into an oatmeal preparation and can be easily transformed into a rice paste with boiled water and can become our everyday breakfast. As a dietary formula, HGMX should be safe and show improved compliance with long-term intake.
However, the efficacy, safety, and compliance of HGMX are based solely on scientific theories and hypotheses, which are still under inspection. Therefore, we plan to conduct a postmarket, hospital-based, randomized controlled trial (RCT) to investigate the long-term efficacy and safety of HGMX on SQD in patients with NOGDs.
This study is a multicenter, randomized, double-blinded, parallel-group, placebo-controlled trial that aims to determine whether HGMX is an effective and safe treatment for improving the symptoms of SQD in patients with NOGDs. The study population will be randomly divided into the HGMX group and the placebo group. The overall design is presented in Figure 1. The trial has been registered at Clinicaltrials.gov (No. NCT03019042). In order to guarantee standardized processes, precise judgment, and the safety of all participants, our study will be performed in accordance with the principles of Good Clinical Practices and the Declaration of Helsinki. We designed and reported our protocol according to Standard Protocol Items: Recommendations for Interventional Trials. .
Figure 1: Overall design of the study.
The potential participants will be recruited in three ways: oral promotion by researchers, hospital posters, and web page publicity. Patients with NOGDs will be recruited from five tertiary hospitals: First Affiliated Hospital of Nanchang University, Second Affiliated Hospital of Nanchang University, Jiangxi Hospital of Traditional Chinese Medicine, Nanchang Hospital of Integrated Traditional Chinese and Western Medicine, and Nanchang Hongdu Hospital of Traditional Chinese Medicine. Subjects who meet the inclusion criteria will be required to sign the informed consent form to show an overall understanding of study details prior to study participation.
We used computer-based statistical analysis software to generate a randomization system that is saved as enciphered data. For allocation concealment, participants are grouped with a centralized randomization assignment. Except in cases of a severe adverse event (SAE), all the participants, medical care providers, data collectors, and outcome reviewers will be blinded before the data analysis. Hence, we established a randomization management center that is in charge of all processes related to random sequence generation, allocation concealment, and blinding. As a result, all the members of this center were excluded from any other processes in this study.
Our participants are grouped into the HGMX group (experimental group) and the control group. In the HGMX group, participants will be treated with an oral dose of 30 g/day HGMX. Every 30 g of HGMX contains 10.1 g of herbal material (detailed components of HGMX are shown in Table 1) and 19.9g early rice and oats. Participants in the control group will receive the same dose of placebo, which consists of 66.7% early rice and 33.3% oats with the same preparation, packaging, labeling, and appearance but a slightly different taste and smell than the real HGMX. All subjects will be treated for a duration of 2 years (104 weeks). During follow-up, patients can be treated as necessary based on clinical guidelines for acute and chronic diseases (such as hypertension, diabetes, and osteoarthritis) that are not included in the exclusion criteria.
Table 1: Detailed components of Hou Gu Mi Xi.
The primary outcome is the change in the score of the Spleen Qi Deficiency Symptoms Grading and Quantifying Scale (SQD scale), which was derived from a classic scale in the Clinical Guidelines of New Drugs for Traditional Chinese Medicine to Validate Spleen Qi Deficiency Symptoms . The SQD scale consists of 17 items that assess (1) stomach distension, (2) abdominal distension, (3) physical fatigue and weakness, (4) mental fatigue and taciturnity, (5) loss of appetite, (6) abnormal stools, (7) stomach pain, (8) stomach tightness, (9) abdominal pain, (10) acid reflux, (11) belching, (12) nausea and vomiting, (13) abnormal bowel sounds, (14) powerless defecation, (15) sallow complexion, (16) loss of taste and hypodipsia, and (17) facial and limb edema. Items 1-6 focus on primary symptoms, and the other items focus on secondary symptoms. The scale assesses all the symptoms of SQD, including the duration of each episode (0-3 points, available for items 1-2 and 7-9), the severity (0-3 points, not available for item 13), and the number of episodes per day (0-4 points, except 0-3 for item 5) and per week (0-5 points). The score of each item will be used to calculate the total score, with 2 points for primary symptoms and 1 point for secondary symptoms. As a result, the total score ranges from 0 to 292, and higher scores indicate more severe symptoms. Compared with the original scale, we have added four symptoms (items 1, 7, 10, and 11) and defined 5 primary symptoms. In addition, we assess not only the severity, which is included in the former scale, but also the duration and the number of episodes per day and per week for these symptoms. For more details, see Table S1 in the supplementary material.
(i)Incidence of adverse events (AEs), including abnormal ECG and dramatic changes (indicated by a value outside 2 × the lower or upper normal reference interval) in alanine transaminase (U/L), aspartate aminotransferase (U/L), total bilirubin (μmol/L), direct bilirubin (μmol/L), indirect bilirubin (μmol/L), or serum creatinine (μmol/L). In addition, doctor-observed and patient-reported AEs are also included.(ii)SAEs are those AEs that could result in treatment in the hospital, a longer in-hospital stay, transfer to an intensive care unit, disability, life-threatening complications, and death.(iii)Treatment-related AEs defined as those AEs fully meet the following condition : (1) the occurrence time is logically related to treatment administration; (2) the adverse effects cannot be interpreted by cointerventions and comorbidities; (3) the adverse effects should relieve or disappear when the treatment is withdrawn; (4) the relation is supported by definitively pharmacological or phenomenological mechanism.(iv)Withdrawal was due to AEs.
The durations of follow-up and treatment are equal at 2 years. In our study, all participants will be evaluated 8 times: at baseline and in weeks 2, 4, 8, 26, 52, 78, and 104. After the qualified participants sign the informed consent form, they will document the characteristics of their own symptoms in the baseline case report form (CRF) and then record the baseline (first visit) physical and laboratory examination data. In weeks 2 and 4, participants will complete the Spleen Qi Deficiency Symptoms Grading and Quantifying Scale (SQD scale) and a physical examination. In weeks 8, 26, 52, 78, and 104, an ECG and laboratory examination, including routine blood, urine, and stool tests, are required. Gastrin-17 and Helicobacter pylori will be assessed at both weeks 52 and 104. Gastroscopy will be performed at only baseline and the last visit. Table 2 provides an overview of the study procedures.
Table 2: Procedures of the trial.
This RCT is a superiority study. From an overall consideration, we will restrict the type I error (α) and type II error (β) to 0.05 and 0.2, respectively. The variation between groups and the combined standard deviation of the primary outcome were estimated at 6.0 and 13.0, respectively, based on the results of a preliminary observational study. In addition, a maximum rate of 20% withdrawal or loss to follow-up was adopted. Therefore, with the corresponding sample size estimation equation  and the consideration of funding constraints, we computed an expected sample size of 200, with 100 in each group.
We will conduct an interim analysis at week 52 and a final analysis at week 104 with the help of the statistical software SPSS v23.0.
Specifically, an intention-to-treat analysis will be adopted for primary outcomes, while the per-protocol analysis will be performed as a sensitivity analysis. The last observation carried forward method will be used for missing data due to loss to follow-up. Continuous and categorical variables will be analyzed by t-test (or analysis of variance) and Pearson chi-squared test, respectively. In addition, continuous variables with a skewed distribution will be compared with the nonparametric rank-sum test. A P value ≤ 0.05 will indicate statistical significance for all the comparisons mentioned above, and group sequential method will be used to adjust the significant boundary for multiple t-test . Subgroup analysis stratified by type of NOGDs (nonatrophic gastritis versus functional gastrointestinal disorders) and infection of Helicobacter pylori (yes versus no) will be performed. Furthermore, an analysis based on safety data set with no imputation will also be conducted. The incidence of AEs in the two groups will be compared using the Pearson chi-squared test, and all the AEs will be described in detail.
Our data collection and management will be carried out under collaboration with our research associates, coordinators, and doctors. All the investigators will undergo standard operating procedures training, in which the follow-up, data collection and management should be strictly performed. The procedures and results of the SQD scale scores, vital sign measurements, sample collection, and laboratory examinations will be performed and obtained under the control and supervision of doctors.
Our research coordinators are college student volunteers from Jiangxi University of Traditional Chinese Medicine who have the responsibilities of contacting patients, helping patients and doctors communicate, and checking data and cross-entry. Meanwhile, several research associates in charge of standard performance, patient compliance, completeness of data collection, and problem solving will be selected from researchers at Jiangxi University of Traditional Chinese Medicine. We will ensure that each study center has a research associate.
The confidentiality of primary CRFs will be protected after data entry. The computerized database will be saved with double password protection and exclusively used for data analysis, in which patient information will be processed anonymously. Only statistical analysis staff will have access to our database, and they will be excluded from procedures other than data analysis.
To enhance patient compliance, the following measures will be adopted: (1) we will fully assess patients' compliance at the stage of eligibility screening to exclude patients with the consideration of poor compliance; (2) we will provide a traffic subsidy (100 China Yuan) to the participants for each follow-up. (3) A 3-day deviation in follow-up will be accepted at weeks 2, 4, and 8, and a 7-day deviation will be acceptable at weeks 26, 52, 78, and 104. (4) We will establish a WeChat (a smartphone application) group to keep in online touch with the patients and provide health counselling services to improve patients' enthusiasm. (5) For all samples allocated to patients, a recycling check will be conducted to ensure patient compliance and to confirm the correct intake dose.
Since the 20th century, the disease spectrum has undergone significant changes: the chronic diseases, such as nonorganic disorders, diabetes mellitus, cardiovascular diseases, and cancer, become the major disease burden. The development of these diseases is multifactorial and the diet often plays an important role. Thus, diet-related management (e.g., diet control and nutrition supplement) could be a treatment approach for these diseases. However, the clinical nutriology simply based on adjustment of diet structure has limitations on the efficacy.
Using the TCM theory, the food (including herbs in the list of food) can be made into dietary TCM formula, which can be daily eaten and has treatment effects for chronic diseases. HGMX is a dietary TCM formula developed based on this concept. After screening of a large number of classic TCM formula, Shen Ling Bai Zhu San was chosen as the basic formula of HGMX, because its efficacy and safety for gastrointestinal diseases are validated by thousands of years of application and animal and clinical researches. On the other hand, except for atractylodes, all components of Shen Ling Bai Zhu San are in the official list of food designated by the Ministry of Health of China. Thus, it can be easily modified to a pure dietary formula on the premise without changing the main effects. During the two years of marketing, no adverse events related to HGMX are reported. Therefore, the safety of this RCT can also be guaranteed.
To validate the potential effects of HGMX on gastrointestinal protection and systemic symptom improvement, we intend to perform a series of RCTs. The study series contains 3 RCTs, with the other two focusing on patients with peptic ulcers (NCT03320538) and on patients who underwent radical gastrectomy for gastric cancer (NCT03025152). These RCTs are also corresponding to three main applications of dietary TCM formula: (1) use for the treatment of nonorganic disorders; (2) adjuvant therapy for the treatment of chronic diseases; and (3) adjuvant therapy for the rehabilitation of major diseases.
The main purpose of this RCT is to clarify the long-term efficacy, safety, and compliance of HGMX, as a dietary TCM formula, in patients with SQD and NOGDs, with the guidance of TCM theory and evidence-based medicine methodology. The randomization of this RCT is standardly designed to minimize the selection bias and the period of follow-up is enough to observe the long-term efficacy and safety. Before the beginning of the trial, we appointed a training group contained senior methodologists, pharmacists, and physicians to conduct a standard operation procedure training in each research hospital. The diagnosis of TCM syndrome and the measurement of patient-reported outcomes will be performed by well-trained TCM physicians according to the standards to increase the consistency between difference investigators and hospitals. We also prespecified two sets of subgroup analysis based on expert's opinion and biological mechanism and used the group sequential method in repeated measures to reduce the probability of type I error. The results of the trial will be reported in strict accordance with the checklist of Consolidated Standards of Reporting Trials (CONSORT) Extension for Chinese Herbal Medicine Formulas. Therefore, this RCT can be expected to produce high-quality evidence.
This study may have some limitations. First, we will use an adjusted scale to assess SQD symptoms without reliability and validity tests, which could introduce a detection bias. We will conduct reliability and validity tests based on our study outcome data to test whether there exist any defects in reliability and validity. Second, because the placebo consists of early rice and oats and will be undertaken regularly according the study protocol, the patients in the placebo group may correct some bad diet habits during the participation, which may have positive effects on NOGDs and lead to bias in results (i.e., the estimate may tend to be conservative). Third, although we will take various measures to improve the patient compliance, the loss to follow-up is still probably inevitable during the long-term follow-up, resulting in attrition bias.
The dietary TCM formula is currently becoming a hot concern in the area of TCM owing to the safety attribute of diet and the therapy attribute of herb. However, the incomplete industry standards and evaluation methodology are restraining the development and application of the dietary TCM formula. A critical reason of these problems is the extreme scarcity of high-quality evidence. Therefore, using rigorous methodology to evaluate efficacy and safety of the dietary TCM formula in the framework of evidence-based medicine is a necessary approach to address these issues. We hope to take the first step to produce high-quality evidence for the dietary TCM formula by carrying out this RCT series, which will establish a reference for the future clinical researches of other dietary TCM formulas and will be of important value in developing related regulations.
This clinical trial project has been approved by Ethics Committee in the First Affiliated Hospital of Nanchang University (2016048), Second Affiliated Hospital of Nanchang University (2016-11), Affiliated Hospital of Jiangxi University of Traditional Chinese Medicine (JZFLL2016093009), Nanchang Hospital of Integrated Traditional Chinese and Western Medicine (2016001), and Nanchang Hongdu Hospital of Traditional Chinese Medicine (KY-2016-001).
Xiaofan Chen and Heyun Nie contributed equally to this work and should be considered as co-first authors. Xiaofan Chen and Heyun Nie designed the study and drafted the manuscript; Wenjun Liu provided methodological advice and critically revised the manuscript; Jianhua Nie and Xu Zhou performed data analyses and drafted the manuscript; Bin Xie, Dongping Chen, Zhangyang Zhao, Yiping Jiang, Kunhe Zhang, Ying Fu, Deping Yang, and Yan Xiong involved the recruitment and follow-up; Weifeng Zhu and Xin Sun conceived and designed the study, developed statistical analysis plan, reviewed the manuscript, and acts as guarantors. All authors have read and approved the final version of the manuscript.
We thank the student volunteers from Jiangxi University of Traditional Chinese Medicine (Gen Den, Yingyu Yue, Xugang Fan, Anting Liao, Fang Yan, Xingxing Yu, and Menghui Wu) for their great efforts on the enrollment of the study. The trial has obtained project approval from Health and Family Planning Commission of Jiangxi Province (no. 2016B026). The trial is financially supported by Scientific Research Funding of Jiangxi Jiangzhong Pharmaceutical (group) Co., Ltd. (no. 2016918). The company will provide HGMX and placebo but will not play any role in study design, conduct, data management, data analysis, and publication.
X. Zheng, Clinical Guideline of New Drugs for Traditional Chinese Medicine, Medicine Science and Technology Press of China, Beijing, China, 1993.
"Tai Ping Hui Min He Ji Ju. Tai Ping Hui Min He Ji Ju Fang, Juan 3 (Imperial Grace Formulary of the Tai Ping Era, Volume 3). Song Dynasty (1102 AD)".


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Ancient History, Galleries and Museums, History, News May 1, 2013
Ginkgo: A Few Amazing Facts
Perhaps the world's most distinctive tree, ginkgo has remained stubbornly unchanged for more than two hundred million years. A living link to the age of dinosaurs, it survived the great ice ages as a relic in China, but it earned its reprieve when people first found it useful about a thousand years ago. Today ginkgo is beloved for the elegance of its leaves, prized for its edible nuts, and revered for its longevity. Ginkgo: The Tree that Time Forgot tells the full and fascinating story of a tree that people saved from extinction – a story that offers hope for other botanical biographies that are still being written.
Today marks the launch of Ginkgo: The Tree That Time Forgot by Peter Crane at Kew Gardens, to celebrate, find a few amazing facts about this peculiar botanical oddity below.
Ginkgo Fossil
Ginkgo is a botanical oddity. Like the platypus among animals, it is a single peculiar species with no close living relatives.
The iconic fan-shaped leaves of ginkgo have been identified as fossils from every continent. Living ginkgo trees have changed little from those that lived 200 million years ago.
Almost driven to extinction by climate change, wild ginkgo trees survive only in China. But you can see ginkgo today in San Francisco's Golden Gate Park, on the streets of Manhattan, and in parks and gardens in all but the warmest and coldest places on our planet.
In the East, Ginkgo has been cultivated for a thousand years for its edible seeds, and many ancient trees are greatly revered by followers of Buddhism, Confucianism, and Shintoism.
The oldest living ginkgo trees in Europe date from 1730 – 1750, when ginkgo was introduced from the East. The oldest ginkgo in North America was planted in the 1780s in the Pennsylvania garden of John Bartram a prominent early American botanist.
Today ginkgo is grown as a botanical curiosity and a resilient street tree; extracts from ginkgo leaves have also become a top-selling herbal medicine that is believed to improve memory and learning. Its efficacy however, remains controversial.
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