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\title[Hook length property of $d$-complete posets via $q$-integrals
(extended abstract)]{Hook length property of $d$-complete posets via
$q$-integrals (extended abstract)}

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\author{Jang Soo Kim\thanks{\href{mailto:jangsookim@skku.edu}{jangsookim@skku.edu}. Jang Soo Kim was supported by NRF grants \#2016R1D1A1A09917506 and \#2016R1A5A1008055.}\addressmark{1} \and Meesue Yoo\thanks{\href{mailto:meesue.yoo@skku.edu}{meesue.yoo@skku.edu}. Meesue Yoo was supported by NRF grants \#2016R1A5A1008055 and \#2017R1C1B2005653.}\addressmark{2}}

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\address{\addressmark{1}Department of Mathematics, Sungkyunkwan University, Korea\\ \addressmark{2}Applied Algebra and Optimization Research Center, Sungkyunkwan University, Korea}

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\received{Nov. 13, 2017}

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\revised{Feb. 28, 2018}

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\abstract{
The hook length formula for $d$-complete posets states that the $P$-partition generating function for them is given by a product in terms of hook lengths. We give a new proof of the hook length formula of $d$-complete posets using $q$-integrals. Proctor proved that any connected $d$-complete poset can be uniquely decomposed into irreducible $d$-complete posets and classified all irreducible $d$-complete posets. In this work, we prove the hook length property of all the irreducible $d$-complete posets. 
The proof is done by a case-by-case analysis consisting of two steps. First, we express the $P$-partition generating function for each case as a $q$-integral and then we evaluate the $q$-integrals. %Several $q$-integrals are evaluated using partial fraction expansion identities and the others are verified by computer. 
}

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%\resume{\lipsum[2]}

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\keywords{Hook length formula, $d$-complete poset, $P$-partition, $q$-integral}

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\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

%----------------------------------------------------------------------------------------------------------------------
\section{Introduction}
%----------------------------------------------------------------------------------------------------------------------


The classical hook length formula due to Frame, Robinson and Thrall \cite{Frame1954} states that for a partition $\lambda$ of $n$, the number $f^{\lambda}$ of standard Young tableaux of shape $\lambda$ is given by
\begin{equation*}
f^\lambda = \frac{n!}{\prod_{x\in \lambda} h(x)},  
\end{equation*}
where $h(x)$ is the hook length of the cell $x$ in $\lambda$. One can naturally consider the shape $\lambda$ as a poset $P$ on the cells in $\lambda$. Then the $P$-partition generating function for the poset also has the following hook length formula:
\begin{equation*}
\sum_{\sigma:P\to\NN}q^{|\sigma|}=\prod_{x\in P}\frac1{1-q^{h(x)}},  
\end{equation*}
where the sum is over all $P$-partitions $\sigma$. 
It is also well known that the $P$-partition generating functions for the posets coming from shifted shapes and forests satisfy the hook length property. 

Proctor \cite{Proctor1999a} introduced $d$-complete posets, which include the posets of shapes, shifted shapes and forests, and 
with Peterson's help, he \cite{Proctor2014} proved that the $d$-complete posets have the hook length property:


\begin{thm}[Hook Length Formula for $d$-complete posets] \label{thm:HLP}
For any $d$-complete poset $P$, we have
\[
\sum_{\sigma:P\to\NN}q^{|\sigma|}=\prod_{x\in P}\frac1{1-q^{h(x)}},
\]
where the sum is over all $P$-partitions $\sigma$.
\end{thm}
We note that Theorem~\ref{thm:HLP} was also proved by Nakada \cite{Nakada2009} and generalized by Ishikawa and Tagawa \cite{IshikawaTagawa07, IshikawaTagawa} to ``leaf posets''. However, their proofs are only sketched in conference proceedings, and so a completely detailed proof of the hook length formula (Theorem~\ref{thm:HLP}) has not been available in the literature.  In this work, we provide a new and complete proof of Theorem~\ref{thm:HLP} using $q$-integrals. This is an extended abstract of \cite{KimYoo2017}. 



%----------------------------------------------------------------------------------------------------------------------
\section{Preliminaries}
%----------------------------------------------------------------------------------------------------------------------


\subsection{Basic definitions and notation} %------------------------------------------------------------



We will use the following notation for $q$-series:
\[
(a;q)_n = (1-a)(1-aq)\cdots(1-aq^{n-1}),
\qquad (a_1,a_2,\dots,a_k;q)_n=(a_1;q)_n \cdots (a_k;q)_n.
\]
Let $\delta_n$ denote the staircase partition $(n-1,n-2,\dots,1,0)$. For a partition $\lambda=(\lambda_1,\dots,\lambda_n)$, the \emph{alternant} $a_\lambda(x_1,\dots,x_n)$ is defined by
\[
a_\lambda(x_1,\dots,x_n) = \det(x_i^{\lambda_j})_{i,j=1}^n.
\]

Given a partition $\lambda$, the \emph{Young diagram} of $\lambda$ is the left-justified array of squares in which there are $\lambda_i$ squares in the $i$th row from the top and the \emph{Young poset} of $\lambda$ is the poset whose elements are the squares in the Young diagram of $\lambda$ with relation $x\le y$ if $x$ is weakly below and weakly to the right of $y$.

If $\lambda$ has no nonzero identical parts, $\lambda$ is called \emph{strict}.
For a strict partition $\lambda$, the \emph{shifted Young diagram} of $\lambda$ is 
the diagram obtained from the Young diagram of $\lambda$ by shifting the $i$th row to the right by $i-1$ units. The \emph{shifted Young poset} of $\lambda$ is defined similarly.
If there is no confusion, we identify a partition $\lambda$ with its Young diagram and also with its Young poset. For a strict partition $\lambda$, the shifted Young diagram of $\lambda$ is denoted by $\lambda^*$. Similarly, the shifted Young poset of $\lambda$ will also be written as $\lambda^*$. 

For a Young diagram or a shifted Young diagram $\lambda$, a \emph{semistandard Young tableau} of shape $\lambda$ is a filling of $\lambda$ with nonnegative integers such that the integers are weakly increasing in each row and strictly increasing in each column.  A \emph{reverse plane partition} of shape $\lambda$ is a filling of $\lambda$ with nonnegative integers such that the integers are weakly increasing in each row and each column.  We denote by $\SSYT(\lambda)$ and $\RPP(\lambda)$ the set of semistandard Young tableaux of shape $\lambda$
and the set of reverse plane partitions of shape $\lambda$, respectively. 

Let $\lambda$ be a strict partition. For $T\in \SSYT(\lambda^*)$ or $T\in\RPP(\lambda^*)$,
the leftmost entry in each row is called a \emph{diagonal entry}. 
We define the \emph{reverse diagonal sequence} $\rdiag(T)$ to be the sequence of diagonal entries in the non-increasing order. 

Now we recall basic properties of $P$-partitions. Let $P$  be a poset with $n$ elements.  A \emph{$P$-partition} is a map $\sigma:P\to\NN$ such that $x\le_P y$ implies $\sigma(x)\ge \sigma(y)$. In other words, a $P$-partition is just an order-reversing map from $P$ to $\NN$. 

For an integer $m\ge0$, we denote by $\PP_{\ge m}(P)$ the set of all $P$-partitions $\sigma$ with $\min(\sigma)\ge m$.  We also define $\PP(P)=\PP_{\ge0}(P)$.  For a $P$-partition $\sigma$, the \emph{size} $|\sigma|$ of $\sigma$ is defined by 
\[
 |\sigma| = \sum_{x\in P}\sigma(x).  
\]

For a poset $P$, we define $\GF(P)$ to be the $P$-partition generating function:
\[
\GF(P) = \sum_{\sigma\in \PP(P)}q^{|\sigma|}.
\]

The following definitions allow us to build $d$-complete posets starting from a chain.

\begin{defn}
Let $P$ be a poset containing a chain $C=\{x_1<x_2<\dots<x_n\}$. 
For $\lambda\in\Par_n$, we denote by $D(P,C,\lambda)$ the poset obtained by taking the disjoint union
of $P$ and $(\lambda+\delta_{n+1})^*$ and identifying $x_n,x_{n-1},\dots,x_1$ with the diagonal elements of 
$(\lambda+\delta_{n+1})^*$. 
\end{defn}

\begin{defn}\label{defn:P(X)}
Let $n$ and $k$ be positive integers. Let
\[
X = \{(\lambda^{(i)}, n_i, s_i): 1\le i\le k\},
\]
where $n_i$ and $s_i$ are positive integers with $s_i+n_i-1\le n$, $\lambda^{(i)}\in\Par_{n_i}$. 
We define $P_{n}(X)$ to be the poset constructed as follows. 
Let $P_0$ be a chain $x_1<x_2<\dots <x_n$ with $n$ elements, called \emph{diagonal entries}. For $1\le i\le k$, we define
$P_i=D(P_{i-1},C_i,\lambda^{(i)})$ where $C_i=\{x_{s_i}<x_{s_i+1}<\dots<x_{s_i+n_i-1}\}$.
Finally we define $P_{n}(X)=P_k$. We also define $P_n^m(X)$ to be the poset obtained from $P_n(X)$ by attaching a chain with $m$ elements above $x_n$. 
We say that an element $y\in P_n (X)$ is \emph{of level} $i$ if $y\le x_i$ and $y\not\le x_{i-1}$.
\end{defn}

Here, we do not provide the detailed definition of $d$-complete posets. In this paper, we basically follow the set up of \cite{Proctor1999}.


%---------------------------------------------------------------------------------------------------------------------------
\subsection{Some properties of $P$-partitions}
%---------------------------------------------------------------------------------------------------------------------------


  For a poset $P$,  let $P^+$ be the poset obtained from $P$ by adding a new element which is greater than all elements in $P$.
If $P$ has a unique maximal element, we define $P^-$ to be the poset obtained from $P$ by removing the maximal element.
Note that  $(P^+)^-=P$ for any poset $P$. If $P$ has a unique maximal element, $(P^-)^+=P$.
There is a simple relation between $\GF(P^+)$ and $\GF(P)$.


\begin{lem}\label{lem:P+}
For a poset $P$ with $p$ elements, we have
\[
\GF(P^+) = \frac{1}{1-q^{p+1}}\GF(P).
\]
\end{lem}

Let $P$ be a poset in which there is a unique maximal element $y_1$ 
and a specified element $y_2$ covered by $y_1$. For integers $m,k\ge 1$, we define $D_{m,k}(P)$
to be the poset obtained from $P$ by adding a disjoint chain $z_m>\dots>z_1>z_0>z_{-1}>\dots>z_{-k}$
and a new element $y_0$ with additional covering relations 
$z_1>y_0, z_0>y_1, z_{-1}>y_2$ and $y_0>y_1$. See Figure~\ref{fig:dmkp}.
We also define $D_{k}(P)$ to be the poset obtained from $D_{m,k}(P)$ by removing the elements
$z_m,\dots,z_1$ and $y_0$.


\begin{figure}[h]
  \centering
\includegraphics{dmk.pdf}  \quad \quad \quad \includegraphics{dmk2.pdf}  
  \caption{The posets $D_{m,k}(P)$ on the left and $D_{k}(P)$ on the right.}
  \label{fig:dmkp}
\end{figure}

Then the following lemma enables us to decompose the $P$-partition generating function of $d$-complete posets.

\begin{lem}\label{lem:dmkp}
Let $P=\{y_1,y_2,\dots,y_p\}$ be a poset in which $y_1$ is the unique maximal element and $y_2$ is covered by $y_1$.
Then
\[
\GF(D_{m,k}(P)) = \frac{1}{(q^{p+k+1};q)_{m+2}}
\left(
\frac{q^{p+1}}{(q;q)_{k-1}} \GF(P^+) + (1-q^{2p+2k+2})\GF(D_{k}(P))
\right).
\]
\end{lem}




\subsection{Semi-irreducible $d$-complete posets}


\begin{defn}\label{def:semiirr}
A $d$-complete poset $P$ is \emph{semi-irreducible} if it is obtained from an irreducible $d$-complete poset by attaching a chain with arbitrary number of elements (possibly 0) below each acyclic element.
\end{defn}

The semi-irreducibility is a slight generalization of the irreducibility defined by Proctor \cite{Proctor1999}.


\begin{lem}\label{lem:attach}
  Let $P_0$ be an irreducible $d$-complete poset with $k$ acyclic elements $y_1,\dots,y_k$.  Suppose that $P_{1},\dots, P_{k}$ are (possibly empty) connected $d$-complete posets having the hook length property. Let $P$ be the poset obtained from $P_0$ by attaching $P_i$ below $y_i$ 
for each $1\le i\le k$, i.e.,
\[
P = (\cdots(P_0 \slant{y_1}{v_1} P_1) \slant{y_2}{v_2} P_2)\cdots \slant{y_k}{v_k} P_k),
\]
where $v_i$ is the unique maximal element of $P_i$. Then $P$ also has the hook length property.
\end{lem}

This lemma tells us that it suffices to prove the hook length property of the semi-irreducible posets to prove that every $d$-complete poset has the hook length property.
Hence we prove:



\begin{thm}\label{thm:semi-irreducible}
Every semi-irreducible $d$-complete poset has the hook length property.  
\end{thm}




%----------------------------------------------------------------------------------------------------------------------
\section{$q$-integrals}
%----------------------------------------------------------------------------------------------------------------------

In this section we express the $P$-partition generating function for $P_n(X)$ as a $q$-integral, where $P_n(X)$ is the poset defined in Definition~\ref{defn:P(X)}.

The \emph{$q$-integral} of a function $f(x)$ over $[a,b]$ is defined by
\[
\int_a^b f(x) d_qx = (1-q)\sum_{i=0}^\infty 
\left( f(bq^i) bq^i - f(aq^i) aq^i \right),
\]
where it is assumed that $0<q<1$ and the sum absolutely converges.

For a multivariable function $f(x_1,\dots,x_n)$ and a partition $\lambda=(\lambda_1,\dots,\lambda_n)$, we denote
$f(q^\lambda) = f(q^{\lambda_1},\dots,q^{\lambda_n})$. 

We define the multivariate $q$-integral over the simplex $\{(x_1,\dots,x_n): 0\le x_1\le\cdots\le x_n\le1\}$ by
\[
\int_{0\le x_1\le\cdots\le x_n\le1} f(x_1,\dots,x_n)\dqx
=\int_{0}^{1}\int_{0}^{x_n} \int_{0}^{x_{n-1}} \dots \int_0^{x_2}f(x_1,\dots,x_n)\dqx.
\]

\begin{lem}\label{lem:par}
We have 
\[
\int_{0\le x_1\le\cdots\le x_n\le1}
f(x_1,\dots,x_n)\dqx=
(1-q)^n \sum_{\mu\in\Par_n} q^{|\mu|} f(q^{\mu}),
\]
where $\Par_n$ denotes the set of partitions of length at most $n$. 
\end{lem}


Note that every semi-irreducible $d$-complete poset can be written as $P_n^m(X)$, by its construction in Definition \ref{def:semiirr}.
By Lemma~\ref{lem:P+}, we have
\begin{equation*}
\GF(P_n^m(X))= \frac{1}{(q^{|P_n(X)|+1};q)_m} \GF(P_n(X)).  
\end{equation*}
We introduce some lemmas which allow us to write  $\GF(P_n(X))$ as a $q$-integral.


\begin{lem}\label{lem:ssyt1}
Let $n$ and $k$ be positive integers and
\[
X = \{(\lambda^{(i)}, n_i, s_i): 1\le i\le k\},
\]
where $n_i$ and $s_i$ are positive integers with $s_i+n_i-1\le n$ and $\lambda^{(i)}\in\Par_{n_i}$. 
For $\mu=(\mu_1,\dots,\mu_n)\in\Par_n$, let
$
\mu^{[i]} = (\mu_{s_i},\mu_{s_{i}+1},\dots,\mu_{s_i+n_i-1}).
$
Then we have
\[
\GF(P_n(X)) = q^{-\sum_{i=1}^n(n-i)\ell_i}\sum_{\substack{\mu\in \Par_n\\ \mu: \rm{strict}}} q^{|\mu|}
\prod_{i=1}^k
\sum_{\substack{T\in \SSYT ( (\delta_{n_i+1}+\lambda^{(i)})^* ) \\
\rdiag(T)=\mu^{[i]}}}  q^{|T|-|\mu^{[i]}|},
\]
where $\ell_i$ is the number of elements of level $i$ in $P_n(X)$.
\end{lem}





\begin{lem}
\label{lem:ssyt2}
For $\lambda,\mu\in\Par_n$ we have
\[
\sum_{\substack{T\in \SSYT ( (\delta_{n+1}+\lambda)^* ) \\
\rdiag(T)=\mu}}  q^{|T|-|\mu|}
=  \frac{(-1)^{\binom n2} a_{\lambda+\delta_n}(q^{\mu}) }
{\prod_{j=1}^{n}(q;q)_{\lambda_j+n-j}}.
\]
\end{lem}

The following result is the key ingredient to express $\GF(P_n(X))$ as a $q$-integral.


\begin{thm}\label{thm:attach}
Let $n$ and $k$ be positive integers and
\[
X = \{(\lambda^{(i)}, n_i, s_i): 1\le i\le k\},
\]
where $n_i$ and $s_i$ are positive integers with $s_i+n_i-1\le n$, $\lambda^{(i)}$ is a partition with $n_i$ parts. 
Suppose that for every $1\le j\le n-1$, there is $1\le i\le k-1$ such that
$s_i\le j<j+1\le s_i+n_i-1$. Then
\begin{multline*}
\GF(P_n(X)) \\= \frac{q^{-\sum_{i=1}^n(n-i)\ell_i}}{(1-q)^{n}}
\int_{0\le x_1\le \cdots\le x_n\le 1}\prod_{i=1}^k
\frac{(-1)^{\binom{n_i}2} a_{\lambda^{(i)}+\delta_{n_i}}(x_{s_i},x_{s_i+1},\dots,x_{s_i+n_i-1})}
{\prod_{j=1}^{n_i}(q;q)_{\lambda^{(i)}_j+n_i-j}} \dqx,
\end{multline*}
where $\ell_i$ is the number of elements of level $i$ in $P_n(X)$.
\end{thm}



%----------------------------------------------------------------------------------------------------------------------
\section{Evaluation of the $q$-integrals}
%----------------------------------------------------------------------------------------------------------------------


In \cite[Table 1]{Proctor1999}, Proctor classified all irreducible $d$-complete posets in $15$ classes, and in \cite{KimYoo2017} slightly generalized 
posets have been considered, namely, semi-irreducible $d$-complete posets. 
In a nutshell, the computation of $q$-integrals corresponding to the $P$-partition generating functions can be summarized as follows. 

\begin{center}
\begin{tabular}{c|l}
\hline
$\qquad$ Classes $\qquad$ & $\qquad$Diagnosis$\qquad$\\
\hline\hline
$1,2$ & shape and shifted shape; proofs are known\\
\hline
$3, 5, 6, 7, 8', 9, 13, 14, 15$ & finite type; can be verified by Sage \cite{sagemath}\\
\hline
$8$-($4$), $10$, $12$ & finite type; but modification is necessary to verify by Sage\\
\hline
$4$, $11$ & infinite type; proof is done by using partial fraction identities\\
\hline
\end{tabular}
\end{center}

In \cite{KimYoo2017}, the class $8$ is divided into $4$ subclasses and $8'$ in the above table includes $8$-($1$),  $8$-($2$) and  $8$-($3$). 
Note that \emph{finite (infinite, resp.) type} means that there are finite (infinite, resp.) number of integration variables in the $q$-integral. 

Here, we demonstrate the computation of one class in each category. 


%-------------------------------------------------------------------------------------------------------
\subsection{Class $2$: Shifted shapes} 
%-------------------------------------------------------------------------------------------------------


\begin{figure}[h]
  \centering
\includegraphics{fig2.pdf}  
  \caption{A semi-irreducible $d$-complete poset of class $2$. This is irreducible if and only if $\mu_1=\mu_2$.}
  \label{fig:class2}
\end{figure}

A semi-irreducible $d$-complete poset of class $2$ is $P_n(X_2)$, where $n\ge4$ and $X_2 = \{(\mu,n,1)\}$, 
with $\mu\in\Par_n$. For $1\le i\le n$, we have $\ell_i = \mu_{n+1-i}+i$.
By Theorem~\ref{thm:attach}, 
\[
\GF(P_n(X_2)) = \frac{q^{-\binom {n+1}3 -\sum_{i=1}^{n-1}i \mu_{i+1}}}{(1-q)^n}
\int_{0\le x_1\le \cdots \le x_n\le 1} 
\frac{a_{\mu+\delta_n}(x_1,\dots,x_n)}{\prod_{i=1}^n(q;q)_{\mu_i+n-i}}
d_qx_1 \cdots d_qx_n.
\]
The hook length property for class 2 is equivalent to
\begin{multline*}
\int_{0\le x_1\le \cdots \le x_n\le 1} a_{\mu+\delta_n}(x_1,\dots,x_n) d_qx_1 \cdots d_qx_n \\
= q^{\binom {n+1}3 +\sum_{i=1}^{n-1}i \mu_{i+1}}(1-q)^n
\frac{\prod_{1\le i<j\le n} (1-q^{\mu_i -\mu_j +j-i})}
{\prod_{1\le i\le j\le n}(1-q^{2n+1-i-j+\mu_i +\mu_{j+1}})},
\end{multline*}
which is proved in \cite[Theorem 8.16]{KimStanton17} using the connection between reverse plane partitions and $q$-integrals. 


%-------------------------------------------------------------------------------------------------------
\subsection{Class $5$: Tailed insets} 
%-------------------------------------------------------------------------------------------------------



A semi-irreducible $d$-complete poset of class $5$ is $P_3 ^{\lambda_1 +1}(X_5)$ for $\lambda\in\Par_2$, $\mu\in\Par_3$ and
$$X_5 =\{ (\lambda,2,1), (\mu,3,1),(\emptyset,2,2),((1),1,1)\},$$
with $\ell_1=\lambda_2 +\mu_3 +2$, $\ell_2 = \lambda_1 +\mu_2+ 3$ and $\ell_3 =\mu_1+4$. 
\begin{figure}[h]
  \centering
\includegraphics{fig5.pdf}  
   \caption{A semi-irreducible $d$-complete poset of class $5$. This is irreducible if and only if $\mu_1=\mu_2$.}
  \label{fig:class5}
\end{figure}

By Lemma~\ref{lem:P+},
$$\GF(P_3 ^{\lambda_1 +1}(X_5))=\frac{1}{(q^{|\lambda|+|\mu|+10};q)_{\lambda_1 +1}}\GF (P_3 (X_5)),
$$
where
\begin{multline*}
\GF (P_3 (X_5))= \frac{q^{-(\sum_{i=1}^2 i(\lambda_i +\mu_{i+1})+7)}}{(1-q)^3}
\int_{0\le x_1\le x_2\le x_3\le 1 } \frac{-a_{\lambda+\delta_2} (x_1,x_2)}{(q;q)_{\lambda_1 +1}(q;q)_{\lambda_2}}\\
\times \frac{-a_{\mu+\delta_3} (x_1,x_2,x_3)}{\prod_{j=1}^3 (q;q)_{\mu_j +3-j}}
\cdot \frac{-a_{\delta_2}(x_2,x_3)}{1-q}\cdot \frac{a_{(1)+\delta_1}(x_1)}{1-q} d_q x_1 d_q x_2 d_q x_3.
\end{multline*}
Then the hook length property for class $5$ is equivalent to the following identity
\begin{multline*}
\int_{0\le x_1\le x_2\le x_3\le 1 } x_1 a_{\delta_2}(x_2,x_3)
a_{\lambda+\delta_2} (x_1,x_2) a_{\mu+\delta_3} (x_1,x_2,x_3)
d_q x_1 d_q x_2 d_q x_3\\
=\frac{(-1) q^{\sum_{i=1}^2 i(\lambda_i +\mu_{i+1})+7}(1-q)^4 (1-q^{\lambda_1-\lambda_2 +1})(1-q^{|\lambda|+|\mu|+\lambda_1 +10})(1-q^{|\lambda|+|\mu|+\lambda_2+9})}{1-q^{|\lambda|+|\mu|+9}}.\\
\times\frac{\prod_{1\le i<j \le 3}  (1-q^{\mu_i -\mu_j +j-i})}{\prod_{i=1}^2 \prod_{j=1}^3 (1-q^{\lambda_i +\mu_j +7-i-j})\prod_{i=1}^3 (1-q^{|\lambda|+|\mu|-\mu_i+4+i})}.
\end{multline*}
This formula has been verified by Sage\cite{sagemath}.


%-------------------------------------------------------------------------------------------------------
\subsection{Class $10$: Tagged Swivels} 
%-------------------------------------------------------------------------------------------------------



A semi-irreducible $d$-complete poset of class $10$ is $P_6 ^{\lambda_1 +4}(X_{10})$ with
$$X_{10}=\{ (\lambda,5,1),(\emptyset ,2,1),((1),2,2),(\emptyset, 2,3),(\emptyset, 3,4),(\emptyset, 2,5) \},$$
where $\lambda\in\Par_5$ and $\ell_1=\lambda_5 +1$, $\ell_2 = \lambda_4 +3$, $\ell_3 = \lambda_3 + 5$, $\ell_4 = \lambda_2 + 5$, $\ell_5 = \lambda_1 +6$, $\ell_6 = 4$. 

\begin{figure}[h]
  \centering
\includegraphics{fig10.pdf}  
   \caption{A semi-irreducible $d$-complete poset of class $10$. This poset is always irreducible.}
  \label{fig:class10}
\end{figure}

To evaluate the $q$-integral, for the sake of the simplicity 
of the computation, we decompose the poset $P_6 ^{\lambda_1 +4}(X_{10})$ using Lemma \ref{lem:dmkp}.


Let $Q=P_5 (X)^-$ for $X=\{ (\mu,5,1), ((1),1,2)\}$ and $\mu=\lambda+(1^5)$.
The poset $P_6 ^{\lambda_1 +4}(X_{10})$ can be also expressed as $D_{\mu_1 +4 ,1}(Q)$
and, by Lemma~\ref{lem:dmkp}, the $P$-partition generating function satisfies the relation  
$$\GF (D_{\mu_1 +4 ,1}(Q))=
\frac{1}{(q^{|\mu|+17};q)_{\mu_1 +6}}(q^{|\mu|+16}\GF (Q ^+) + (1-q^{2|\mu|+34})\GF (D_1 (Q))).$$
Note that $Q^+=P_5 (X)$ and $D_1 (Q)=P_5 (X')$ where 
$X'=\{ (\mu,5,1), ((1),1,2), (\emptyset, 2,4)\}$. 
By Theorem~\ref{thm:attach}, 
\[
\GF (Q ^+) = \frac{q^{-\sum_{i=1}^5 (i-1)\mu_{i}-23}}{(1-q)^6\prod_{i=1}^5 (q;q)_{\mu_i +5-i}}
\int_{0\le x_1 \le \cdots \le x_5 \le 1} x_2 a_{\mu+\delta_5 }(x_1,\dots, x_5) d_q x_1 \cdots d_q x_5
\]
and
\begin{multline*}
 \GF (D_1 (Q)) \\= \frac{(-1)q^{-\sum_{i=1}^5 (i-1)\mu_{i}-23}}{(1-q)^7\prod_{i=1}^5 (q;q)_{\mu_i +5-i}}
\int_{0\le x_1 \le \cdots \le x_5 \le 1} x_2 (x_4 -x_5)a_{\mu+\delta_5 }(x_1,\dots, x_5) d_q x_1 \cdots d_q x_5.
\end{multline*}

The above $q$-integrals with $4$ variables can be explicitly computed by computer and 
the hook lengths of the elements in $P_6 ^{\lambda_1 +4}(X_{10})=D_{\mu_1+4,1}(Q)$ can be explicitly computed.
By combining the aforementioned observations, we obtain that the hook length property for class 10 is equivalent to
\begin{multline*}
\prod_{i=1}^5 \frac{1-q^{|\mu|+\mu_i-i+23}}{  (1-q^{\mu_i +6-i})(1-q^{|\mu|-\mu_i +10+i})}
\prod_{1\le i<j\le 5}\frac{1-q^{\mu_i-\mu_j+j-i}}{1-q^{\mu_i+\mu_j-i-j+13}}\\
=\sum_{\ell=1}^5\frac{(-1)^{5-\ell}q^{-\sum_{i=1}^5 (i-1)\mu_{i}-23}}{(1-q)^5 (1-q^{|\mu|+16})}
 \left( (1-q^{2|\mu|+23}) g(\widehat{\mu}^{(\ell)},0) 
-(q^{|\mu|+16};q)_2 \cdot g(\widehat{\mu}^{(\ell)},1) \right),
\end{multline*}
where 
\begin{multline*}
g(\nu,m):=\int_{0\le x_1\le \cdots \le x_4 \le 1} x_2 x_4^m a_{\mu +\delta_4 }(x_1,x_2,x_3,x_4) d_q x_1\cdots d_q x_4\\
 =  \frac{q^{12+\sum_{i=1}^4 i\mu_{i+1}}(1-q)^4 (1-q^{|\mu| +12})\prod_{1\le i<j\le 4}(1-q^{\mu_i -\mu_j+j-i})}{(1-q^{|\mu|+11+m})\prod_{1\le i<j\le 4}(1-q^{\mu_i +\mu_j +11-i-j})\prod_{i=1}^4 (1-q^{\mu_i+5-i})},
\end{multline*}
for $\nu\in\Par_5$ and an integer $m\ge0$.
We have verified this identity by computer.




%-------------------------------------------------------------------------------------------------------
\subsection{Class $4$: Insets} 
%-------------------------------------------------------------------------------------------------------
\begin{figure}[h]
  \centering
\includegraphics{fig4.pdf}  
  \caption{A semi-irreducible $d$-complete poset of class $4$. This is irreducible if and only if $k=0$ and $\mu_1=\mu_2$.}
  \label{fig:class4}
\end{figure}

A semi-irreducible $d$-complete poset of class $4$ is $P_{n+1} ^m(X_4)$, where $n\ge2$, $k\ge0$ and
$$X_4= \{(\lambda, n-1, 1), (\mu, n+1,1), ((k),1,n)\},
$$
for  $\lambda\in \Par_{n-1}$ and $\mu\in\Par_{n+1}$. 
In this poset, 
$\ell_j = \lambda_{n-j}+\mu_{n-j+2}+2j-1$ for $1\le j\le n-1$, $\ell_n = \mu_2 +n+k$ and $\ell_{n+1}=\mu_1 + n+1$. 


By applying Lemma~\ref{lem:P+} and Theorem~\ref{thm:attach}, we obtain
$$\GF(P_{n+1} ^{\lambda_1 +n-2}(X_4))=\frac{1}{(q^{|\lambda|+|\mu|+n^2 +k+3};q)_{\lambda_1 +n-2}}\GF (P_{n+1}(X_4)),
$$
where
\begin{multline*}
 \GF (P_{n+1}(X_4)) = \frac{q^{-(\sum_{i=1}^n ((i+1)\lambda_{i}+i\mu_{i+1})+\frac{1}{6}n(n-1)(2n+5)+1+k)}}{(1-q)^{n+1}}
 \int_{0\le x_1\le \cdots \le x_{n+1}\le 1}\frac{a_{(k)}(x_n)}{(q;q)_k} \\
\times \frac{(-1)^{\binom{n-1}{2}}a_{\lambda+\delta_{n-1}}(x_1,\dots, x_{n-1})}{\prod_{i=1}^{n-1}(q;q)_{\lambda_i +n-1-i}}
\cdot \frac{(-1)^{\binom{n+1}{2}}a_{\mu +\delta_{n+1}}(x_1,\dots, x_{n+1})}{\prod_{i=1}^{n+1}(q;q)_{\mu_i +n+1-i}}
 d_q x_1\cdots d_q x_{n+1}.
\end{multline*}

Taking the explicit hook lengths of the elements in the poset $P_{n+1}^{\lambda_1 +n-2}(X_4)$ into consideration,
the hook length property for class $4$ can be written as the following identity
\begin{multline*}
 \int_{0\le x_1 \le \cdots \le x_{n+1}\le 1} x_n ^k a_{\lambda +\delta_{n-1}}(x_1,\dots, x_{n-1}) a_{\mu +\delta_{n+1}}(x_1,\dots, x_{n+1}) d_q x_1 \cdots d_q x_{n+1}\\
= \frac{(-1)q^{\sum_{i=1}^n ((i+1)\lambda_{i}+i\mu_{i+1})+\frac{1}{6}n(n-1)(2n+5)+1+k}(1-q)^{n+1}}{\prod_{i=1}^{n+1}(1-q^{|\lambda|+|\mu|-\mu_i+n(n-1)+k+i})}\cdot \frac{\prod_{j=1}^{n-1}(1-q^{|\lambda|+|\mu|+\lambda_j +n^2 +n-j+k+1})}{1-q^{|\lambda|+|\mu|+n^2+k+2}}\\
\quad \times \frac{\prod_{1\le i<j\le n+1}(1-q^{\mu_i-\mu_j+j-i})\prod_{1\le i<j\le n-1}(1-q^{\lambda_i -\lambda_j +j-i})}{\prod_{\substack{1\le i\le n+1\\ 1\le j \le n-1}}(1-q^{\mu_i+\lambda_j +2n-i-j+1})},
\end{multline*}
or 
\begin{multline*}
\frac{\prod_{j=1}^{n-1}(1-q^{|\lambda|+|\mu|+\lambda_j +n^2 +n-j+k+1})}{\prod_{i=1}^{n+1}(1-q^{|\lambda|+|\mu|-\mu_i +n^2 -n+k+i})}\\
= \sum_{\ell=1}^{n+1}\frac{q^{-|\lambda|-|\mu|+\mu_\ell-n^2+n-k-\ell}}{1-q^{|\lambda|+|\mu|-\mu_\ell+n^2-n+k+\ell}}
\cdot\frac{\prod_{j=1}^{n-1}(1-q^{\mu_\ell+\lambda_j+2n-\ell-j+1})}{\prod_{j=1,j\ne \ell}^{n+1}(1-q^{\mu_\ell-\mu_j+j-\ell})}.
\end{multline*}
This identity can be proved by applying a partial fraction expansion
identity \cite[p.~451]{WW}
\begin{equation*}
\frac{\prod_{j=1}^{n+1}(1-b_j/t)}{\prod_{j=1}^n (1-a_j/t)}=\sum_{\ell=1}^n 
\frac{\prod_{j=1}^{n+1}(1-a_\ell /b_j)}{(1-a_\ell/t)\prod_{j=1,j\ne \ell}^{n}(1-a_\ell/a_j)},
\quad \text{ for } b_1\cdots b_{n+1} =a_1\cdots a_n t,
\end{equation*}
by making appropriate substitutions for $a_i$'s, $b_i$'s and $t$. 
We omit the details.














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\begin{equation}\label{eqn:eq1}
\int_0^\infty \exp(-x)dx = 1.
\end{equation}

\begin{lem}\label{lem:lem1}
\lipsum[11]
\end{lem}

\lipsum[10]

\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{plot}
\caption{A plot of a function.}
\label{fig:plot}
\end{figure}

\subsection{A subsection}

\lipsum[5]

\begin{align*}
1 + 1 + 1 &= 2 + 1 \\ &= 3.
\end{align*}

\subsubsection{A subsubsection}

\lipsum[17]

Here is a citation~\cite{greenwade93} with URL. Here is another
citation~\cite{MR4}, which is the earliest-numbered Math Review with a
functioning DOI: MR0000004. Finally, a reference to
equation~\eqref{eqn:eq1}. And a reference to Figure~\ref{fig:plot}.

\acknowledgements{\lipsum[20]}

