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\title[Naruse hook formula for linear extensions of mobile posets]{Naruse hook formula for linear\\ extensions of mobile posets}

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\author[GaYee Park]{GaYee Park\thanks{\href{mailto:park@math.umass.edu}{park@math.umass.edu} received support form the NSF grant (DMS-1855536).}\addressmark{1}}

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\address{\addressmark{1}Department of Mathematics and Statistics, University of Massachusetts, Amherst}

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\abstract{
Linear extensions of posets are important objects in enumerative and algebraic combinatorics that are difficult to count in general. Families of posets like straight shapes and $d$-complete posets have hook-length product formulas to count linear extensions whereas families like skew shapes have determinant or positive sum formulas like the Naruse hook length formula from 2014. In 2020, Garver et. al. gave determinant formulas to count linear extensions of a family of posets called mobile posets that refine $d$-complete posets and border strip skew shapes. We give a Naruse type hook length formula to count linear extensions of such posets as well as $q$-analogues of our formula in both major and inversion index.}

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\keywords{standard tableaux, skew shapes, hook formula}

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\begin{document}

\maketitle
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\section{Introduction}

The number of standard Young tableaux (SYT) of a shape $\lambda$ is counted by the famous hook-length formula:

\begin{theorem}[{Frame-Robinson-Thrall \cite{FRT}}]
Let $\lambda$ be a partition of $n$. We have
    \begin{equation*}
       |\SYT(\lambda)| =  n!\prod_{u \in [\lambda]}\frac{1}{h(u)},
    \end{equation*}
where $h(u) = \lambda_i + \lambda_j' -i - j +1$ is the hook length of the square $u= (i,j)$.
\end{theorem}

Naruse introduced a more general formula to count the number of SYT of skew-shape as a sum over excited diagrams of products of hook-lengths.

\begin{theorem}[{Naruse \cite{naruse2014}}]
For a skew shape $\lambda/\mu$ of size $n$, we have
\begin{equation}
    |\SYT(\lambda/\mu)| \,=\, n!\sum_{D \in \mathcal{E}(\lambda/\mu)} \prod_{u\in [\lambda] \setminus D}\frac{1}{h(u)}, \tag{NHLF}
\end{equation}
where $\E(\lambda/\mu)$ is the set of excited diagrams of $\lambda/\mu$.
\end{theorem}

In \cite{MPP2}, Morales, Pak and Panova introduces a combinatorial proof of NHLF based on the case of \emph{border strips}, connected skew-shapes with no $2\times 2$ box.\\

The number of SYT of shape $\lambda/\mu$ can also be interpreted as the number of linear extension of a poset induced by the Young diagram of $\lambda/\mu$. In \cite{Pro}, Proctor  defined the family of \emph{d-complete} posets, that include Young diagrams and rooted trees, and proved a hook-length formula to count the number of linear extensions.

\begin{theorem}[{Peterson-Proctor \cite{Pro}}]\label{thm: proctor}
The number of linear extensions of a $d$-complete poset $\mathcal{P}$ with $n$ element is
\[e(\mathcal{P}) = \frac{n!}{\prod_{x\in \mathcal{P}}h_{\mathcal{P}}(x)},\]
where $h_{\PP}(x)$ is the hook-length of $x\in \PP$.
\end{theorem}

A \emph{mobile poset} is a recent common refinement of border strips and $d$-complete posets introduced in \cite{GMM} (see Figure~\ref{fig: poset to mobile}, (a)). The authors found a determinantal formula for the number of linear extensions of these posets, similar to \emph{Jacobi--Trudi} formula and asked whether there was a Naruse-type formula \cite[Sec. 6.1]{GMM} for this number. The first main result of this extended abstract is a Naruse hook-length formula for mobile posets. 

\begin{theorem}[NHLF for mobiles]\label{thm: mobile nhlf}
Let $P_{\lambda/\mu}(\mathbf{p})$ be a free-standing mobile poset of size $n$ with underlying border strip $\lambda/\mu$ and $\mathbf{p} = (p_{(r_1,s_1)},\dots, p_{(r_k,s_k)})$ the $d$-complete posets hanging on $(r,s)$. Then we have, 
\begin{equation} \label{eq: main thm}
    e(P_{\lambda/\mu}({\bf p})) = \frac{n!}{H({\bf p})} \sum_{D\in \E(\lambda/\mu)} \prod_{(i,j)\in \gamma} \frac{1}{h'(i,j)},
\end{equation}
where $h'(i,j) =  \lambda_i - i + \lambda'_j-j + 1 + \sum_{r\geq i,s\geq j}|p_{({r,s})}|$, and $H(\mathbf{p})$ is the product of hook lengths of all elements in the $d$-complete posets hanging from $\lambda/\mu$.
\end{theorem}

As an application to this main theorem, we give bounds to generalizations of Euler number defined in \cite{GMM}. See Corollary~\ref{cor: boundary} and Corollary~\ref{cor: zigzag bound}.



Moreover, there is the following $q$-analogue of the NHLF for semistandard Young tableaux proved in \cite{MPP1}. We state this result in terms of $e_q^{\maj}(P,\omega):=\sum_{\sigma} q^{\maj(\sigma)}$ where $(P,\omega)$ is a labeled poset, and $\sigma$ is a linear extension of it.


\begin{theorem}[{Morales-Pak-Panova \cite{MPP1}}]
For a skew shape $\lambda/\mu$ with associated poset $Q_{\lambda/\mu}$ with $\omega$ Schur labeling, we have:
\begin{equation}
   \frac{e_q^{\maj}(Q_{\lambda/\mu},\omega)}{\prod^n_{i=1}(1-q^i)}  = \sum_{D\in \mathcal{E}(\lambda/\mu)} q^{w(Br(D))} \prod_{u \in [\lambda]\setminus D} \frac{1}{1-q^{h(u)}}
\end{equation}
where $w(Br(D))$ is the sum of hook-lengths of the support of broken diagonals.
\end{theorem}

Naruse-Okada \cite{NaruseOkada} have a different $q$-analogue of $e_q^{\maj}(P,\omega)$ for a family called \emph{skew $d$-complete} posets with \emph{natural labelings}.

Our second result is the $q$-analgoue of Theorem~\ref{thm: mobile nhlf} in terms of major index for all mobile posets.

\begin{theorem}\label{conj: major mobile nhlf}
For a labeled mobile poset $(P_{\lambda/\mu},\omega)$ with $\omega$ reverse Schur labeling on $[\lambda/\mu]$ and natural labeling on $d$-complete posets,
\begin{equation}\label{eq: major mobile nhlf}
    \frac{e_q^{\maj}(P_{\lambda/\mu},\omega)}{\prod^n_{i=1}(1-q^i)} = \prod_{v\in \mathbf{p}}\frac{1}{1-q^{h(v)}}\sum_{D\in \mathcal{E}(\lambda/\mu)} q^{w'(\Br(D))} \prod_{u\in [\lambda]\setminus D} \frac{1}{1-q^{h'(u)}},
\end{equation}
     where $w'(\Br(D)) = \sum_{u\in \Br(D)} h'(u)$.
\end{theorem}


We also have a $q$-analogue in terms of inversion statistic for the \emph{mobile trees} ($d$-complete posets that are restricted to rooted trees). 


\begin{theorem}\label{thm: inv nhlf}
For a labeled mobile poset $(P_{\lambda/\mu},\omega)$ with $\omega$ reverse Schur labeling on $[\lambda/\mu]$ and natural labeling on $d$-complete posets,
    \begin{equation}\label{thm: inv q-analogue}
   \frac{e_q^{\inv}(P_{\lambda/\mu},\omega)}{\prod^n_{i=1}(1-q^i)} = \prod_{v\in \mathbf{p}}\frac{1}{1-q^{h(v)}}\sum_{D\in \mathcal{E}(\lambda/\mu)} q^{w(Br(D))+ q^{p_{D}}} \prod_{u\in [\lambda]\setminus D} \frac{1}{1-q^{h'(u)}}, \end{equation}
     where $w(Br(D)) = \sum_{u\in Br(D)} h(u)$ and $p_D = \sum_{(i,j)\in [\mu]\setminus D}\sum_{b=j} p_{a,b}$.
\end{theorem}


% As a partial progress, we give a new proof of  $e_q^{\inv}(P_{\lambda/\mu},\omega)=e_q^{\maj}(P_{\lambda/\mu},\omega)$ for a border strip $P_{\lambda/\mu}$ using Foata's \cite{F} classical bijection on permutations.



In Section~\ref{sec: background}, we give definitions and background results required for the proof. In Section~\ref{sec: nhlf proof}, we give an example and the proof for Theorem~\ref{thm: mobile nhlf}. In Section~\ref{sec: application}, we show an application to the main theorem. Finally, In Section~\ref{sec: q-analogue}, we give examples and sketch the proofs of the $q$-analogues.


The full length of this extended abstract is available at \cite{Park}.

\section{Background} \label{sec: background}

\subsection{Posets and linear extensions}

A \emph{linear extension} of an $n$-element poset $\PP$ is a bijection $f: \PP \to [n]$ that is order-preserving. The number of SYT of a shape $\lambda/\mu$ is equal to the number of \emph{linear extensions} of a poset of shape $\lambda/\mu$. We denote  the set of linear extensions of $\PP$ as $\mathcal{L}(\PP)$, and $e(\PP) = |\mathcal{L}(\PP)|$. We have the following formula for the number of linear extensions of a disjoint sum. 

\begin{proposition}\label{prop: disjoint sum}
The number of linear extensions of a disjoint sum of posets $\PP_i$, each of size $n_i$, is $e(\PP_1+ \cdots +\PP_k) = \binom{ n_1+ \cdots n_k}{n_1, \cdots n_k} e(\PP_1)\cdots e(\PP_k)$.
\end{proposition}



\subsection{Border strips and Mobile Posets}
A \emph{border strip} is a connected skew shape $\lambda/\mu$ containing no $2\times 2$ box. A family of \emph{d-complete posets} are a large class of posets containing rooted tree posets and posets arising from Young diagrams. Given a border strip, we can convert it into a poset by letting the inner corners of the diagram be the maximal points of the corresponding poset. Now we can construct a mobile poset (See Figure~\ref{fig: poset to mobile} (a) for an example).

\begin{definition}[{Garver-Grosser-Matherne-Morales \cite{GMM}}]
A \emph{mobile\footnote{What we call a mobile poset is called a free-standing mobile poset in \cite{GMM}.} (tree) poset} is a poset obtained from a border strip $Q$, by allowing every element $x \in Q$ to cover the maximal element of a nonnegative number of disjoint  $d$-complete (rooted tree) posets.
\end{definition}

\begin{figure}
    \centering
    \subfigure[]{\includegraphics[width=4.5cm]{figures/tableau_to_poset.pdf}}
    \hspace{1cm}
    \subfigure[]{\includegraphics[height=.8cm]{figures/ExciedMove.pdf}}
    \hspace{.5cm}
    \subfigure[]{\includegraphics[height=1.5cm]{figures/EDexample.pdf}}
    \caption{(a) The conversion of tableaux to mobile posets. (b) Excited move (c) Excited diagrams and the corresponding broken diagonals of $(2,2,2,1)/(1,1)$}
    \label{fig: poset to mobile}
\end{figure}

\subsection{Excited diagrams and broken diagonals}

Denote $[\lambda/\mu]$ as the skew shape Young diagram of a shape $\lambda/\mu$. An \emph{excited diagram} of $\lambda/\mu$, denoted by $D$, is a subset of $[\lambda]$ obtained from $\mu$ by applying a sequence of \emph{excited moves} that we define next. Let $D\in \E(\lambda/\mu)$, then $(i,j) \in D$ is an \emph{active cell} if $(i+1,j), (i,j+1)$, and $(i+1,j+1)$ are not in $D$. We obtain a new excited diagram by replacing an active cell by $(i+1,i+j)$ (see Figure~\ref{fig: poset to mobile}, (b)). Note that for border strips, the excited diagrams can also be interpreted as the complement of its lattice paths $\gamma$ from $(\lambda_1',1)\to (1,\lambda_1)$ that stay inside $[\lambda]$ (see\cite[Sec. 3]{MPP2}).

For each excited diagram $D\in \E(\lambda/\mu)$ we associate a set of \emph{broken diagonals} $Br(D) \subset [\lambda]\setminus D$ as follows. Start with $D = [\mu]$, then $\Br(D) = \{(i,j)\in\lambda/\mu | i-j = \mu_t-t\}$, for $1 \leq t \leq \ell(\lambda)-1$ and $\mu_t=0$ if $\ell(\mu)<t\leq \ell(\lambda)$. For each active cell $u=(i,j)$ and its excited move $\alpha_u:D\to D'$, we have a corresponding move for the broken diagonal where $\Br(D') = \Br(D)\setminus \{(i+1,j+1)\} \cup \{ (i+1,j)\}\}$. See Figure \ref{fig: poset to mobile}, (b) and (c).



\subsection{Multivariate function}

For border strip $\lambda/\mu$, let

\begin{equation}{\label{eq: multivariate}}
F_{\lambda/\mu}(\mathbf{x}, \mathbf{y})= F_{\lambda/\mu}(x_1,\dots, x_{\lambda_1'}, y_1,\dots, y_{\lambda_1}) : = \sum_{D\in \E(\lambda/\mu)} \prod_{(i,j)\in [\lambda]\setminus D} \frac{1}{x_i - y_j}.
\end{equation}

Let $\lambda/\nu$ be the shape obtained by removing an inner corner of $\lambda/\mu$. We denote this subtraction by $\mu\to\nu$. Note that for border strips, $\lambda/\nu$ is disconnected. We denote each disconnected component as $\lambda/\nu_1$ and $\lambda/\nu_2$.

We need the following identity of $F_{\lambda/\mu}(\mathbf{x}, \mathbf{y}) $ from \cite{MPP2}.
\begin{lemma}[{Pieri-Chevalley formula \cite[Eq. (6.3)]{MPP2}}] 
\begin{align}\label{eq: Chevalley}
F_{\lambda/\mu} (\mathbf{x},\mathbf{y}) &= \frac{1}{x_1-y_1}\sum_{\mu \to \nu} F_{\lambda/\nu^1}({\bf x},{\bf y}) F_{\lambda/\nu^2}({\bf x},{\bf y}), 
%F(A\to B) &= \frac{1}{x_1 - y_1}\sum_C F(A\to C^*)F(C^* \to B), 
\end{align}
where $\lambda/\nu^1$ and $\lambda/\nu^2$ are the two connected border strips that form $\lambda/\nu$.
\end{lemma}

% \begin{remark}
% The original equation from \cite{MPP2} uses slightly different notation. Translating from their notations, let $A= (\lambda_1',1),B = (1,\lambda_1)$ and $C= (r,s)$. Note that in the border strip, $\lambda/\nu$ is always disconnected unless the removed inner corner is either $A$ or $B$. Then $F(A\to C^*) = F((\lambda_1',1)\to (r+1,s))$ and $F(C^* \to B) = F((r,s+1)\to (1,\lambda_1))$. We denote them $F_{\lambda/\nu^1}$ and $F_{\lambda/\nu^2}$ respectively. 
% \end{remark}


\subsection{$q$-analogues of linear extensions}

A \emph{labeled poset} $(\PP,\omega)$ is a poset $\PP$ with a labeling $\omega:\PP \to [n]$. We call $\omega$ a \emph{natural labeling} if for any $x,y\in \PP$ with $x<_{\PP} y$, we have $\omega(x) < \omega(y)$ \cite{sta12}. We call $\omega$ a \emph{reverse Schur labeling} if the labeling increases as it follows the path from the right end  of the poset \cite{sta12}. For Theorem~\ref{thm: inv nhlf} and Conjecture~\ref{conj: major mobile nhlf}, we use reverse Schur labeling on $[\lambda/\mu]$ and natural labeling on the $d$-complete posets. In the case of inversion statistic, for each $\mu\to\nu$, we need $\omega(x_1)>\omega(x_2)$ for all $x_i \in \lambda/\nu_i$ to satisfy the condition for Proposition~\ref{prop: qanalogue disjoint poset}

Given a linear extension $f: \PP \to [n]$, the permutation $\omega \circ f^{-1} \in \mathfrak{S}_n$ is a \emph{linear extension of the labeled poset}, $\mathcal{L}(\PP,\omega)$. Recall, $\maj(\sigma)= \sum_{i\in \des(\sigma)} i$ and $\inv(\sigma)= \#\{i\in [n-1]| \sigma_i > \sigma_j \mbox{ where } i<j\}$ where $\des(\sigma):= \{i \in [n-1] \,|\, \sigma_i > \sigma_{i+1}\}.$ The two common statistics for $q$-analogues of the number of linear extensions for a labeled poset $(\mathcal{P},\omega)$ are the major index and inversions.


Let  $\mbox{stat}\in \{\maj, \inv\}$, the \emph{major index (inversion) q-analogue} of the number of linear extensions of a labeled poset $(\mathcal{P},\omega)$ is 
\[e_q^{\stat}(\mathcal{P},\omega) := \sum_{\sigma\in \mathcal{L}(\mathcal{P},\omega)}q^{\stat(\sigma)}.
\]


Now we state the $q$-analogue of the hook-length formulas. We define the \emph{q-integer} $[n]_q:= 1+q+\cdots + q^{n-1}$ and the \emph{q-factorial} $[n]_q! := [n]_q\cdots[2]_q[1]_q$. We also define the \emph{$q$-multinomial coefficients} $\scriptsize{\qbin{n}{p}{q}}$ $ := \frac{[n]!}{[p]_q![n-p]_q!}$. 

Peterson and Proctor \cite{Pro} gave a $q$-analogue formula in terms of major index for $d$-complete posets.

\begin{theorem}[{Peterson and Proctor \cite{Pro}}]
Let $(\PP, \omega)$ be a labeled $d$-complete poset of size $n$ with any labeling. Then,
\[e_q^{\maj(\mathcal{P},\omega)} = q^{\maj(\PP,\omega)} \frac{[n]_q!}{\prod_{x\in\PP} [h_{\PP}(x)]_q}.\]
\end{theorem}

Bj\"{o}rner and Wachs \cite{BW} gave a $q$-analogue formula in terms of inversion index for rooted tree posets.

\begin{theorem}[{Bj\"{o}rner and Wachs \cite{BW}}]\label{thm: inv hook length}
Let $(\PP, \omega)$ be a rooted tree poset with a natural labeling. Then,
\[e_q^{\inv}(\mathcal{P},\omega) = q^{\inv(\PP,\omega)} \frac{[n]_q!}{\prod_{x\in\PP} [h_{\PP}(x)]_q}.\]
\end{theorem}

For both major \cite[Exercise 3.162(a)]{sta12} and inversion \cite{BW} statistics, we have the following proposition.

\begin{proposition}\label{prop: qanalogue disjoint poset}
Let $(P+Q,\omega)$ be a labeled disjoint sum of posets with $|P+Q| = n$ and $|P| = p$. Suppose that $\omega$ has the property that the label of every element of $P$ is smaller than the label of every element of $Q$. We have
\[e_q^{\inv}(P+Q,\omega) = \qbin{n}{p}{q} e_q^{\inv}(P,\omega_1)\cdot e_q^{\inv}(Q,\omega_2),\]
where $\omega_1$ and $\omega_2$ are the labeling obtained by restricting $\omega$ to $P$ and $Q$ respectively.
\end{proposition}

The formula for the disjoint sum in major index is the same, and it applies for all labeling.


\section{Proof of the NHLF for Mobiles}\label{sec: nhlf proof}

In this section we sketch the proof of Theorem~\ref{thm: mobile nhlf}. The proof follows the proof of the NHLF for border strips in \cite{MPP2}. We need to first define the hook lengths of mobile posets. Given a mobile poset $P_{\lambda/\mu}(\mathbf{p})$, define the hook length of $(i,j) \in [\lambda]$ as follows:
\begin{equation} \label{eq: def modified hook length}
h'(i,j) = \lambda_i - i + \lambda_j' - j +1 + \sum_{a\geq i,\\ b\geq j} p_{a,b}
\end{equation}


In other words, it is the usual hook length of the cell in $\lambda$ plus the sizes of the $d$-complete posets that are attached on the segment of the border strip inside of the hook of $(i,j)$ (see Figure \ref{fig:mobile example} (a)). We provide an example of the main theorem below.

\begin{example}
Consider the mobile poset $P_{2221/11}$ from Figure~\ref{fig:mobile example} (b). By Theorem~\ref{thm: mobile nhlf}, one can check that
\begin{align}
    e(P) = \frac{13!}{2^4\cdot 4^2}\left(\frac{1}{5\cdot 6\cdot 7^2} + \frac{1}{5\cdot 6 \cdot 7^2 \cdot 12} + \frac{1}{5 \cdot 7^2 \cdot 12 \cdot 13} \right) = 33000.
\end{align}
\begin{figure}[h]
\centering
\subfigure[]{\includegraphics[height = 2.5cm]{figures/poset_to_tableau_mobile.pdf}}
\quad
\subfigure[]{\includegraphics[height=2.5cm]{figures/mobile_ex.pdf}}
\quad
\subfigure[]{\includegraphics[height=2.5cm]{figures/mobile_ex_lable.pdf}}
\quad
\subfigure[]{\includegraphics[height=2.5cm]{figures/euler.pdf}}

    \caption{(a): $h'(u)$ is the usual hook-length plus the size of $d$-complete posets in the shaded area. (b): mobile poset with  \textcolor{blue}{hook-lengths} marked. (c): a \textcolor{red}{labeled} mobile tree. (d): illustration of poset $\mathcal{C}_p(k)$ (top) and $\mathcal{A}_p(k)$ (bottom).}
    \label{fig:mobile example}
\end{figure}
\end{example}


Now we give the proof of the theorem. Let $H_{\lambda/\mu}$ be the sum on the RHS of \eqref{eq: main thm}. Then we have the following lemma.

\begin{lemma} \label{lemma: recurrence in H}
    $H_{\lambda/\mu} =\frac{1}{n} \sum_{\mu\to\nu} H_{\lambda/\nu^1}H_{\lambda/\nu^2}$
\end{lemma}

\begin{proof}
We evaluate $F(\mathbf{x}, \mathbf{y})$ from \eqref{eq: multivariate} at the following values of $x_i$ and $y_j$: 

\[
x_i = \lambda_i-i + 1 - \sum_{a<i} p_{a,b} \quad \mbox{ and } \quad y_j = j-\lambda_j' - \sum_{b\geq j} p_{a,b}.
\]

Note that $x_1-y_1 = n$ and $x_i - y_j = h'(i,j)$. Evaluating $F_{\lambda/\mu}(\mathbf{x},\mathbf{y})$ at such $x_i$ and $y_j$, we obtain the sum on the RHS of \eqref{eq: main thm}. 

\begin{equation} \label{eq: evaluated F}
    \left.F_{\lambda/\mu} (\mathbf{x},\mathbf{y}) \right.\mid_{\substack{x_i = \lambda_i - i + 1 + \sum_{a\geq i} p_{a,b} \\ y_j = j-\lambda'_j + \sum_{b<j}p_{a,b}}} = \sum_{\gamma: (\lambda'_1,1)\to (1, \lambda_1),\gamma \subset \lambda} \prod_{(i,j)\in \gamma} \frac{1}{h'(i,j)} = H_{\lambda/\mu}
\end{equation}

Then, evaluating \eqref{eq: Chevalley} at such $x_i$ and $y_j$, we obtain the desired equation. 
\end{proof}


Similar to skew shaped SYT, we have the following recurrence for mobile posets:

\begin{lemma}\label{lemma:recurrence e(P)}
For a mobile poset $P_{\lambda/\mu}$, we have
\begin{equation}\label{eq:recurrence e(P)}
    e(P_{\lambda/\mu}) = \sum_{\mu\to \nu} e(P_{\lambda/\nu}),
\end{equation}
where $\nu$ is obtained by adding an inner corner of $\lambda/\mu$ to $\mu$.
\end{lemma}




\begin{proof}
For a fixed mobile $P_{\lambda/\mu}$, a linear extension of this poset consists of an  inner corner of $\lambda/\mu$ followed by a linear extension of the remaining poset of shape $\lambda/\nu$, where $\mu \to \nu$. Conversely, given a linear extension of $P_{\lambda/\nu}$, by inserting the new element in the beginning we obtain a linear extension of $P_{\lambda/\mu}$.
\end{proof}


We are now ready to give the proof for Theorem~\ref{thm: mobile nhlf}.


\begin{proof}[Proof of Theorem \ref{thm: mobile nhlf}]
 We induct on $|\lambda/\mu|$ to show that $e(P_{\lambda/\mu}(\mathbf{p}))/n! \,=\, H_{\lambda/\mu}/H(\mathbf{p})$ using Lemma \ref{lemma:recurrence e(P)}. Note that $\lambda/\nu$ is disconnected and
\begin{equation}\label{eq: disconnected}
    P_{\lambda/\nu} =P_{\lambda/\nu^1}+P_{\lambda/\nu^2} +T_1 + \cdots + T_k,
\end{equation}
where $P_{\lambda/\nu^1},P_{\lambda/\nu^2}$ are the two connected free standing mobiles of size $p_1$ and $p_2$, and $T_1,\dots,T_k$, for $k\geq 0$, are the $d$-complete posets of sizes $t_1,\ldots,t_k$ hanging from the removed inner corner in $P_{\lambda/\mu}$.
By Proposition~\ref{prop: disjoint sum} we have  
\begin{equation*} %\label{eq: e multinomial}
e(P_{\lambda/\nu}) = \binom{ n-1}{p_1,p_2,t_1,\dots, t_k} e(P_{\lambda/\nu^1})e(P_{\lambda/\nu^2})e(T_1)\cdots e(T_k),
\end{equation*}
then using this, \eqref{eq:recurrence e(P)} becomes
\begin{equation} \label{eq:recurrence e(P) refined}
\frac{e(P_{\lambda/\mu})}{(n-1)!} \,=\, \sum_{\mu\to \nu}\left(\prod_{i=1}^k \frac{e(T_i)}{t_i!}\right) \frac{e(P_{\lambda/\nu^1})}{p_1!} \frac{e(P_{\lambda/\nu^2})}{p_2!}. 
\end{equation}

Since $T_i$ is $d$-complete then $e(T_i)/t_i! = 1/H(T_i)$, by Thoerem~\ref{thm: proctor}. Also,  both \[|\lambda/\nu^1|,|\lambda/\nu^2| < |\lambda/\mu|,\] by induction $e(P_{\lambda/\nu^j})/{p_j!} = H_{{\lambda/\nu^j}}/H(\mathbf{p_j})$ for $j=1,2$, where $H(\mathbf{p_j})$ is the product of the hook lengths of the elements in the $d$-complete posets hanging from  $[\lambda/\nu^j]$. Note that $H(\mathbf{p})= H(\mathbf{p_1}) H(\mathbf{p_2}) H(T_1)\cdots H(T_k)$. Then \eqref{eq:recurrence e(P) refined} becomes 
\begin{equation} \label{eq: recurrence e(P) hook sum corners}
e(P_{\lambda/\mu}) = \frac{(n-1)!}{H(\mathbf{p})} \sum_{\mu\to \nu}  H_{{\lambda/\nu^1}} H_{\lambda/\nu^2}.
\end{equation}

By Lemma~\ref{lemma: recurrence in H},
the sum on the RHS of \eqref{eq: recurrence e(P) hook sum corners} equals $n\cdot H_{\lambda/\mu}$, completing the proof.
\end{proof}

\section{Application}\label{sec: application}

As an application to Theorem~\ref{thm: mobile nhlf}, it gives bounds to $e(P_{\lambda/\mu}(\mathbf{p}))$ just as in \cite{MPPasymptotics}.

\begin{corollary}\label{cor: boundary}
For any mobile poset $e(P_{\lambda/\mu}(\mathbf{p}))$ of size $n$, 
\[\frac{n!}{H(\mathbf{p})\prod_{u\in [\lambda/\mu]}h'(u)} \leq e(P_{\lambda/\mu}(\mathbf{p})) \leq |\E(\lambda/\mu)|\cdot \frac{n!}{H(\mathbf{p})\prod_{u\in [\lambda/\mu]}h'(u)}\]
where $[\lambda/\mu]$ is the border strip of the mobile poset.
\end{corollary}

One application of the formula is that it provides bounds to generalizations of \emph{Euler numbers} defined in \cite{GMM}. The authors give two generalizations of Euler number using two different families of posets, up-down posets with $k-1$ downs and chains (or anti-chains) of size $p$  hanging on every minimal element,denoted as $\mathcal{C}_p(k)$ and $\mathcal{A}_p(k)$ (see Figure~\ref{fig:mobile example} (d)). See \href{https://oeis.org/A332471}{A332471} and \href{https://oeis.org/A332568}{A332568} in the OEIS for examples of these sequences.

\begin{corollary}\label{cor: zigzag bound}
%For a mobile poset $P_{\lambda/\mu}(\mathbf{p})$ that is either $\mathcal{C}_p(k)$ or $\mathcal{A}_p(k)$ we have

\[\frac{(2k+kp)!}{ (p+1)!^k(2p+3)^{k-1}(p+2)} \leq e(\mathcal{C}_p(k)) \leq 
\Cat(k)\cdot \frac{(2k+ kp)!}{ (p+1)!^k(2p+3)^{k-1}(p+2)}\]

\[\frac{(2k+ kp)!}{(p+1)^k(2p+3)^{k-1}(p+2)} \leq e(\mathcal{A}_p(k)) \leq 
\Cat(k)\cdot \frac{(2k + kp)!}{(p+1)^k(2p+3)^{k-1}(p+2)},\]
where $\mathcal{Z}$ is the up-down border strip with $k-1$ many down steps. 
\end{corollary}

\begin{proof}
The result follows from Corollary~\ref{cor: boundary}, a routine calculations of hooks, and the fact that the excited diagrams of up-down posets are given by the Catalan numbers \cite{MPP2}
\end{proof}



\section{$q$-analogues of the Naruse-type formula for mobiles}\label{sec: q-analogue}

\subsection{An inversion index $q$-analogue}

In this section we give an example and sketch the proof of Theorem~\ref{thm: inv nhlf}. Unless specified otherwise, $(P_{\lambda/\mu}(\mathbf{p}),\omega)$ is a labeled mobile tree poset.

\begin{example}
Consider labeled the mobile tree poset $(P_{2221/11},\omega)$ from Figure~\ref{fig:mobile example} (c). By Theorem~\ref{thm: inv nhlf}, we have

\begin{align*}
    e_q^{\inv}(P) &= \frac{[11]!}{[1]^4[3]^2}\left(\frac{q^4}{[4][6][1][5][6]} + \frac{q^9}{[4][6][10][5][6]} + \frac{q^{14}}{[4][6] [10] [11] [6]} \right)\\
    & = q^{38} + 4 q^{37} + 9 q^{36} + 17 q^{35} +\cdots + 9 q^6 + 4q^5 + q^4.
\end{align*}
\end{example}


% \begin{remark}
% Note that Theorem~\ref{thm: inv nhlf} is only for mobile trees, where the $d$-complete posets are restricted to rooted trees. This is because there is no known hook length formula for $e_q^\inv(\mathcal{P})$ when $\mathcal{P}$ is a $d$-complete poset.
% \end{remark}

Denote the sum on the RHS of \eqref{thm: inv q-analogue} as $\widetilde{H}_{\lambda/\mu}(q)$. We have the following lemma.

\begin{lemma}\label{lem: Chevalley qanalog inv}
    We have $(1-q^n)\cdot \widetilde{H}_{\lambda/\mu} =   \sum_{\mu\to\nu}  q^{\lambda_1'-1 +c(u)} \cdot \widetilde{H}_{\lambda/\nu^1}(q) \cdot \widetilde{H}_{\lambda/\nu^2}(q),$
where $c(u) = j-i$ for $u = (i,j)$, the inner corener from $\mu\to\nu$
\end{lemma}

\begin{proof}
We first evaluate $F_{\lambda/\mu}(\mathbf{x},\mathbf{y})$ at $x_i=q^{\lambda_i - i + 1 - \sum_{a< i} p_{a,b}}$ and $y_j = q^{j-\lambda'_j - \sum_{b\geq j} p_{a,b}}$.
\begin{equation} \label{eq: evaluated F_q mobile}
    \left.F_{\lambda/\mu} (\mathbf{x},\mathbf{y}) \right.\mid_{\substack{x_i = q^{\lambda_i - i + 1 - \sum_{a< i} p_{a,b}},y_j = q^{j-\lambda'_j - \sum_{b\geq j} p_{a,b}}}} = (-1)^n \cdot \sum_{\substack{\gamma: A \to B,\\ \gamma\subset \lambda}}\prod_{(i,j)\in \gamma}\frac{q^{\lambda_j'-j + \sum_{b\geq j} p_{a,b}}}{1-q^{h'(i,j)}}
\end{equation}

By  \cite[Prop 4.7]{MPP1} and \cite[Lemma 7.17]{MPP1}, we have
\begin{align}
    \sum_{(i,j) \in \lambda\setminus D}\Big( (\lambda_j'-j) + \sum_{b\geq j} p_{a,b} \Big) &= \sum_{(i,j) \in [\lambda]\setminus D}\Big( (\lambda_j'-i) + \sum_{b\geq j} p_{a,b} \Big) - \sum_{(i,j)\in [\lambda]\setminus[\mu]} c(i,j) \nonumber \\ 
    &=  w(\Br(D)) + \sum_{(i,j) \in [\lambda]\setminus D} \sum_{b\geq j }p_{a,b} - \sum_{(i,j)\in [\lambda]\setminus[\mu]} c(i,j) \label{eq: broken diagonal inv}
\end{align}

where $c(i,j) = j-i$ and $w(\Br(D)) = \sum_{(i,j)\in Br(D)} h(i,j)$. Denote $\sum_{(i,j) \in [\lambda/\mu]} \sum_{b\geq j }p_{a,b}$ as $p^*$. Taking $p^*$ and $c(i,j)$ outside of the sum, we can write \eqref{eq: evaluated F_q mobile} as
\begin{equation}
      \left.F_{\lambda/\mu} (\mathbf{x}|\mathbf{y}) \right.\mid_{\substack{x_i = q^{\lambda_i - i + 1 - \sum_{a< i} p_{a,b}}, y_j = q^{j-\lambda'_j - \sum_{b\geq j} p_{a,b}}}} = (-1)^n \cdot q^{p^*- \sum_{(i,j)\in [\lambda/\mu]}c(i,j)} \widetilde{H}_{\lambda/\mu}(q)
\end{equation}

Then, evaluating \eqref{eq: Chevalley} at such $x_i$ and $y_j$, we obtain the desired formula.
\end{proof}

We generalize Lemma \ref{lemma:recurrence e(P)} to the inversion index $q$-analogue. We have the following.

\begin{lemma}\label{lem: inv recurrence}
\begin{equation}\label{eq: recurrsion inv}
    e_q^{\inv}(P_{\lambda/\mu},\omega) = \sum_{\mu\to\nu} q^{n-\omega(u)} e_q^{\inv}(P_{\lambda/\nu}, \omega_\nu)
\end{equation}
where $\omega$ is a reverse Schur labeling and $\omega(u)$ is the label of the inner corner $u$ from $\mu\to\nu$.
\end{lemma}

\begin{proof}
Recall that the linear extension $\sigma$ of $P_{\lambda/\mu}$ consist of a linear extension $\sigma'$ of $\lambda/\nu$ followed by an inner corner $u$. Note that $\inv(\sigma) = \inv(\sigma') + n - \omega$, where $n-\omega$ is the number of inversion caused by the inner corner. The rest of the proof follows the same argument as Lemma~\ref{lemma:recurrence e(P)}.
\end{proof}

We are now ready to give the proof for Theorem~\ref{thm: inv nhlf}.

\begin{proof}[Proof of Theorem \ref{thm: inv q-analogue}]

We show that $e_q^{\inv}(P_{\lambda/\mu}(\mathbf{p})) =\frac{\prod_{i=1}^n(1-q^i)}{\prod_{v\in T} 1-q^{h(v)}} \cdot \widetilde{H}_{\lambda/\mu}(q)$ by induction on $|\lambda/\mu|$ using Lemma~\ref{lem: inv recurrence}. Recall $\lambda/\nu$ is disconnected and is expressed as \eqref{eq: disconnected}. By induction and Theorem~\ref{thm: inv hook length}, for each $P_{\lambda/\nu^j}$ and $T_i$, we have
% $\frac{e_q(P_{\lambda/\nu^j})}{[p_j]_q!} = \frac{\prod_{i=1}^{p_j}(1-q^i)}{\prod_{v\in \mathbf{p_j}}1-q^{h(v)}}\cdot \widetilde{H}_{\lambda/\nu^j} \cdot \frac{(1-q)^{p_j}}{\prod_{i=1}^{p_j}(1-q^i)} =\frac{(1-q)^{p_j}\cdot }{\prod_{v\in \mathbf{p_j}} 1-q^{h(v)}}\cdot  \widetilde{H}_{\lambda/\nu^j},$

% $\frac{e_q^{\inv}(T_i)}{[t_i]_q!} = \frac{q^{\inv(T_i)}}{\prod_{v\in T_i}[h(v)]_q} =\frac{(1-q)^{t_i}}{\prod_{v\in T_i}(1-q^{h(v)})}.$

$\frac{e_q(P_{\lambda/\nu^j})}{[p_j]_q!}  =\frac{(1-q)^{p_j}\cdot }{\prod_{v\in \mathbf{p_j}} 1-q^{h(v)}}\cdot  \widetilde{H}_{\lambda/\nu^j}(q) \mbox{     and    }\frac{e_q^{\inv}(T_i)}{[t_i]_q!} = \frac{q^{\inv(T_i)}}{\prod_{v\in T_i}[h(v)]_q} =\frac{(1-q)^{t_i}}{\prod_{v\in T_i}(1-q^{h(v)})}.$

 Note that $T_i$ are natural labeling, so $\inv(T_i)=0$ vanishes. Using Proposition~\ref{prop: qanalogue disjoint poset} and the equations above, we have

\[ e_q(P_{\lambda/\nu}) =\frac{\prod_{i=1}^{n-1}(1-q^i)}{\prod_{v\in T} (1-q^{h(v)})}  \widetilde{H}_{\lambda/\nu^1}(q)\cdot \widetilde{H}_{\lambda/\nu^2}(q) \]

We now apply the equation to \eqref{eq: recurrsion inv}. 
\begin{equation}\label{eq: recurrence substitued}
    e_q(P_{\lambda/\mu}) =\frac{\prod_{i=1}^{n-1}(1-q^i)}{\prod_{v\in T} (1-q^{h(v)})} \sum_{\mu\to \nu}q^{n-\omega(u)}\cdot  \widetilde{H}_{\lambda/\nu^1}(q)\cdot \widetilde{H}_{\lambda/\nu^2}(q)
\end{equation}

Note that $\lambda_1'-1 + c(u) = n-\omega(u)$ for $\omega$, reversed Schur labeling. By Lemma~\ref{lem: Chevalley qanalog inv}, the sum on the RHS of \eqref{eq: recurrence substitued} equals $(1-q^n)\cdot \widetilde{H}_{\lambda/\mu}(q)$, completing the proof.
\end{proof}





\subsection{A major index $q$-analogue}

In this section, we give an example and sketch the proof of Theorem~\ref{conj: major mobile nhlf}. The poset $(P_{\lambda/\mu}(\mathbf{p}),\omega)$ is labeled with a  reversed Schur labeling on the border strip and a natural labeling on the $d$-complete posets.

\begin{example}
Consider  the  poset $(P_{2221/11},\omega)$ from Figure~\ref{fig:mobile example} (c). By Theorem~\ref{conj: major mobile nhlf}, we have
\begin{align*}
    e_q^{\maj}(P) &=  q^{44} + 4q^{43} + 9q^{42}+ 17q^{41} +\cdots + 9q^{12} + 4q^{11} + q^{10}\\
    &= \frac{[11]!}{[1]^4[3]^2}\left(\frac{q^{10}}{[4][6][1][5][6]} + \frac{q^{15}}{[4][6][10][5][6]} + \frac{q^{20}}{[4][6] [10] [11] [6]} \right).
\end{align*}
\end{example}

Similiarly as the case of the inversion index, we have the following lemmas in terms of the major index. Again, denote the sum of the RHS of \eqref{eq: major mobile nhlf} as $H_{\lambda/\mu}(q)$.

\begin{lemma}\label{lemma: Chevalley qanalog maj}
We have $(1-q^n)\cdot H_{\lambda/\mu}(q) =   \sum_{\mu\to\nu} {q^{|P_{\lambda/\nu^1}|}} \cdot H_{\lambda/\nu^1}(q)\cdot H_{\lambda/\nu^2}(q),$
 where $T_\nu$ is the union of the $d$-complete posets that were hanging on the removed inner corner $u$.
\end{lemma}

\begin{proof}[Proof sketch]

As done in the inversion index case, we first evaluate the multivariate formula $F_{\lambda/\mu}$ at $x_i =q^{\lambda_i - i + 1 - \sum_{a< i}} p_{a,b}$ and $y_j = q^{j-\lambda'_j - \sum_{b\geq j} p_{a,b}}$. We modify \eqref{eq: broken diagonal inv} as follows:

\begin{align*}
    \mbox{LHS} &=  w'(\Br(D)) + \Bigg(\sum_{(i,j) \in [\lambda]\setminus D} \sum_{b\geq j }p_{a,b} - \sum_{(i,j)\in \Br(D)}p_{i,j}\Bigg) - \sum_{(i,j)\in [\lambda]\setminus[\mu]} c(i,j)
\end{align*}
where $w'(\Br(D)) = \sum_{(i,j)\in \Br(D)}h'(i,j)$.
Let $p_1$ be the size of the $d$-complete posets on $P_{\lambda/\nu_1}$. Applying this modification to \eqref{eq: Chevalley}, we get the following equation.

\begin{equation}\label{eq: Chevalley qanalog maj}
    (1-q^n)\cdot{H}_{\lambda/\mu}(q) =   \sum_{\mu\to\nu}  q^{\lambda_1'-1 +c(u) + p_1} \cdot {H}_{\lambda/\nu^1}(q)\cdot {H}_{\lambda/\nu^2}(q).
\end{equation}
 Note that $\lambda_1'-1 + c(u) + p_1 = |P_{\lambda/\nu_1}|$.
\end{proof}

\begin{lemma}\label{conj: major mobile recurrence}
    For a labeled mobile poset $(P_{\lambda/\mu}(\mathbf{p}),\omega)$, where $\omega$ is a reverse Schur labeling, 
    $e_q^{\maj}(P_{\lambda/\mu},\omega) = \sum_{\mu\to\nu}q^{|P_{\lambda/\nu_1}|} e_q^{\maj}(P_{\lambda/\nu},\omega_\nu)$
where $\lambda/\nu_1$ is the left disconnected poset of $\lambda/\nu$, and $\omega_\nu$ is the restricted labeling of $\omega$ onto $\lambda/\nu$.
\end{lemma}




% \subsection{Major index $q$-analogue for border strips}

% The recurrence formula for the major index in terms of $\mu \to \nu$, we need a bijection is not trivial. Given $\sigma \in \mathcal{L}(P_{\lambda/\mu},\omega)$ and $\sigma' \in \mathcal{L}(P_{\lambda/\nu}, \omega_\nu)$ as described in Lemma~\ref{lem: inv recurrence} the decrease in the major index from $\sigma$ to $\sigma'$ is not distributed equally. To understand the recurrence in the major index , we need a bijection that switches $\maj$ and $\inv$. We use the Foata's classical bijection (see \cite[Sec. 1.4]{Sta1}), denoted by $\varphi$ defined as follows. Let $w = w_1\cdots w_n \in  \mathfrak{S}_n$, and we define $\gamma_1,\dots \gamma_n$, where $\gamma_k$ is a permutation of $\{w_1,\dots,w_k\}$.

% Let $\gamma_1 = w_1$. For each $k\geq 1$, if the last letter of $\gamma_k$ is greater (respectively smaller) than $w_{k+1}$, then split $\gamma_k$ after each letter greater (respectively smaller) than $w_{k+1}$. To obtain $\gamma_{k+1}$, cyclically shift each compartment of $\gamma_k$ to the right, then place $w_{k+1}$ at the end. We set $\varphi(w) = \gamma_n$

% We have the following bijection between the major and inversion index.

% % \begin{theorem}[Foata \cite{F}]
% % Let $\varphi$ be the Foata bijection. For all $w \in \mathfrak{S}_n$, $\des(w^{-1}) = \des(\varphi(w)^{-1})$.
% % \end{theorem}


% \begin{lemma}
% Given a $\sigma \in \mathcal{L}(P, \omega)$, where $\omega$ is a (reverse) Schur labeling, $\phi(\sigma) \in \mathcal{L}(P,\omega)$ also, and
% $\maj(\sigma) = \inv(\varphi(\sigma))$
% \end{lemma}

% \begin{proof}
% The lemma above is a special case of \cite[Theorem 2.2]{BW}. 
% \end{proof}

% Consequently, we have the following recurrence
% % \begin{lemma}\label{lem: inv = maj}
% % For a border strip $Q_{\lambda/\mu}$,
% % $e_q^{\inv}(Q_{\lambda/\mu},\omega) = e_q^{\maj}(Q_{\lambda/\mu},\omega)$
% % where $\omega$ is a reverse Schur labeling
% % \end{lemma}


% % Then by Lemma~\ref{lem: inv = maj} and Lemma~\ref{lem: inv recurrence}, we have the following recurrence.


% \begin{lemma}\label{lem: maj border strip}
% For a border strip $Q_{\lambda/\mu}$,
% \[e_q^{\maj}(Q_{\lambda/\mu},\omega) = \sum_{\mu\to\nu}q^{|\lambda/\nu_1|} e_q^{\maj}(Q_{\lambda/\nu},\omega_\nu)\]
% where $\lambda/\nu_1$ is the bottom left disconnected piece of $\lambda/\nu$, and $\omega_\nu$ is the restricted labeling of $\omega$ onto $\lambda/\nu$.
% \end{lemma}

% Note that $n-\omega(u) = |\lambda/\nu_1|$ for the case of border strip and when $\omega$ is a (reverse) Schur labeling. Lemma~\ref{lem: maj border strip} resembles our conjecture~\ref{conj: major mobile recurrence}. We hope to understand the recurrence in the mobile case by extending the Foata bijection to the case of mobile posets.


\acknowledgements{I would like to thank Alejandro Morales for introducing this problem and for all of his guidance throughout this project. I would also like to thank Stefan Grosser, Jacob Matherne,  Soichi Okada, and the anonymous referees for helpful comments.}

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