\section{Definitions and Preliminaries}\label{sec:prelim}
\subsection{Strict partitions and shifted tableaux}
A \textit{strict partition} $\lambda$ is a sequence of strictly decreasing positive integers $(\lambda_1>\lambda_2>\cdots>\lambda_d>0)$, where $d$ is the number of (nonzero) parts of $\lambda$. We denote $|\lambda| = \sum_{i = 1}^{d} \lambda_i$ as the \textit{size} of $\lambda$. For a strict partition $\lambda$ its corresponding %\textit{shifted tableaux}, or
\textit{shifted shape}, consists of $\lambda_i$ boxes in row $i$, shifted $d-i+1$ steps to the left. More specifically, the shifted shape is the diagram
\[D(\lambda):=\{(i,j{-}d{+}i{-}1)\:|\: 1\leq i\leq d,\ 1\leq j\leq \lambda_i\}.\]
For simplicity of notation, we also use $\lambda$ to denote its shape $D(\lambda)$. Note that for a shifted shape, its columns $-(d-1),\ldots,0$ form a staircase shape of length $d$ flipped horizontally. For a shifted shape $\lambda$, define a \emph{shifted tableau} $T$ to be a filling of $D(\lambda)$ with non-negative integers. For any shifted tableau $T$, let $\sh(T)$ denote its underlying shifted shape. 

Throughout the paper, we fix the number $d$, that is the length of all the shifted shapes we are going to consider. We also write $\bar i$ to mean $-i$. 

\begin{defin}\label{def:standard}
A shifted tableau $T$ of shape $\lambda$ is called a \emph{standard Young tableau} if it is a filling of $1,2,\ldots,|\lambda|$ that is increasing in rows and columns.
%A \textit{standard Young tableau} of shifted shape $\lambda$ is a filling of $\lambda$ with $1,2,\ldots,|\lambda|$, increasing in rows and columns.
\end{defin}
The set of standard Young tableaux of shape $\lambda$ is denote $\SYT(\lambda)$ and its cardinality is denoted $f^{\lambda}$. The number $f^{\lambda}$ can be computed via the hook length formula as we explain here. For a box $(i,j)\in\lambda$ with $j\geq0$, its \textit{hook} $H(i,j)$ consists of all the boxes in row $i$ to the right of $(i,j)$, all the boxes in column $j$ below $(i,j)$ and the box $(i,j)$ itself. For a box $(i,\bar j)\in\lambda$ with $j>0$, its \textit{hook} $H(i,\bar j)$ consists of all the boxes in row $i$ to the right of $(i,\bar j)$, all the boxes in column $\bar j$ below $(i,\bar j)$, the box $(i,\bar j)$ itself and all the boxes in row $d-j+1$. Let $h(i,j)=|H(i,j)|$ be the size of the hook.
\begin{theorem}\cite{Thrall}
For a shifted shape $\lambda$, $f^{\lambda}=|\lambda|!\,/\prod_{x\in\lambda}h(x).$
\end{theorem}

To define an analogous notion of balanced tableaux, as in \cite{BalancedTableaux}, for shifted shapes, we need some more notions. For a filling $B$ of shape $\lambda$, its \textit{extended filling} $\tilde{B}$ is a filling of the extended shape \[\tilde{\lambda}=\lambda\cup\{(1,\bar d),(2,\overline{d{-}1}),\ldots(d,\bar1)\}\] which agrees with $B$ on $\lambda$ and equals $B(i,0)$ on the newly added box $(i,-(d+1-i))$. The \textit{extended hook} is defined as $\tilde{H}(i,j)=H(i,j)$ for $j\geq0$, and $\tilde{H}(i,\bar j)=H(i,j)\cup\{(d+1-j,\bar j)\}$ for $j>0$. See \cref{ex:BST621} for visualization.

For a box $(i,j)\in\lambda$, we also define its \textit{rank function} $\rk(i,j)$. If $j\geq0$, let $\rk(i,j)$ be the number of boxes in row $i$ of $H(i,j)$, and let $\rk(i,\bar j)$ be $2$ plus the number of boxes in $H(i,j)$ with positive column index. More formally,
\[
\rk(i,j)=\begin{cases}
\lambda_i-d+i-j\ &\text{if }j\geq0,\\
\lambda_i-d+i+\lambda_{d+1+j}+j+1 &\text{if }j<0.
\end{cases}
\]
We can now introduce our main object of study:
\begin{defin}\label{def:balanced}
A shifted tableau $B$ of shape $\lambda$ is called a \textit{balanced shifted tableau} if it is a filling of $1,2,\ldots,|\lambda|$ such that $B(i,j)$ is the $\rk(i,j)$-th largest entry in the extended hook $\tilde H(i,j)$ of $\tilde{B}$ for all $(i,j)\in \lambda$. Define $\BS(\lambda)$ to be the set of balanced shifted tableaux of shape $\lambda$.
\end{defin}

% \begin{remark}
% If we instead naively define $\rk(i,j)$ to be the length of the \textit{right arm} of $H(i,j)$ as in straight shapes, i.e. define the balanced condition to be $B(i,j)$ remains unchanged after reordering the elements in $H(i,j)$ (or $\tilde{H}(i,j)$ ), then the number of such tableaux is different from $f^{\lambda}$.
% \end{remark}

\begin{remark}
We remark that our definition of balanced tableaux is different from the balanced filling in Section 6 of \cite{Hamakerthesis} and the standard $w$-tableau as in \cite{K95}. One can derive a result similar to \cref{thm:main} using their definition, but we note that the two results are fundamentally different. The difference can be interpreted loosely as studying the balanced tableaux of dominant permutation v.s. Grassmannian permutation in the framework of \cite{fomin1997balanced}.
\end{remark}

\begin{ex}\label{ex:BST621}
Let $\lambda=(6,2,1)$ and consider the balanced shifted tableau in \cref{fig:BST621}. The hook $H(1,-1)$ contains the colored boxes so $h(1,-1)=7$, while the extended hook $\tilde{H}(1,-1)$ contains one more box at coordiante $(3,-1)$, which is circled and filled with $1$. As this hook contains $3$ boxes with positive column index, we have $\rk(1,-1)=5$. The balanced condition is now satisfied at coordinate $(1,-1)$ as $3$ is indeed the $5$-th largest numbers among the numbers in the extend hook, $9,5,2,4,3,7,1,1$.
\begin{figure}[h!]
\centering
\ytableausetup{boxsize=1.5em}
\begin{ytableau}
\none[4]&6&*(yellow)3&*(yellow)4&*(yellow)2&*(yellow)5&*(yellow)9\\
\none&\none[8]&*(yellow)7&8&\none&\none&\none\\
\none&\none&\none[\circled{1}]&*(yellow)1&\none&\none&\none
\end{ytableau}
\caption{A balanced shifted tableau of shape $(6,2,1)$}
\label{fig:BST621}
\end{figure}
\end{ex}

\begin{comment}
\begin{remark}
Here is another way to understand the extended hooks $\tilde{H}(i,j)$ and ranks $\rk(i,j)$, shown in \cref{fig:BSTsymmetric}. Given a shifted tableau $B$, we stack its extended filling $\tilde{B}$ and a flipped copy $B^T$ (see left of \cref{fig:BSTsymmetric}) together to obtain a larger tableau $B_0$. The entries in the shaded boxes on the diagonal agree with column $0$ of $B$. Then the extended hook $\tilde{H}(i,j)$ of $x=(i,j)$ is the same as the standard hook of $x$ in $\tilde{B}$ (colored in blue). The rank $\rk(i,j)$ is the number of yellow boxes (see right of \cref{fig:BSTsymmetric}) in the hook of $x$.
\begin{figure}[h!]
\centering
\begin{tikzpicture}[scale = 0.4]
\fill[yellow!100] (-4,4) rectangle (-3,3);
\fill[yellow!100] (-3,3) rectangle (-2,2);
\fill[yellow!100] (-2,2) rectangle (-1,1);
\fill[yellow!100] (-1,1) rectangle (0,0);
% \draw (0,0) -- ++(2,0) --++(0,1) -- ++(1,0) --++(0,1) --++(2,0) --++(0,2) --++(-8,0) --++(0,-1) --++(1,0) --++(0,-1) --++(1,0) --++(0,-1) --++(1,0) -- cycle;
\draw (0,0) -- ++(2,0) --++(0,1) -- ++(1,0) --++(0,1) --++(2,0) --++(0,2) --++(-9,0) --++(0,-1);
\node at (1,2) {$\tilde{B}$};
\draw (0,0) -- ++(0,-2) --++(-1,0) -- ++(0,-1) --++(-1,0) --++(0,-2) --++(-2,0) --++(0,8) --++(1,0) --++(0,-1) --++(1,0) --++(0,-1) --++(1,0) --++(0,-1) -- cycle;
\node at (-2,-1) {$B^T$};
% \draw (-3,4) -- (-4,4) -- (-4,3);

\begin{scope}[xshift=400pt]
\fill[yellow] (0,3) rectangle (5,2);
\fill[yellow] (-2,0) rectangle (-1,-3);
\draw (0,0) -- ++(2,0) --++(0,1) -- ++(1,0) --++(0,1) --++(2,0) --++(0,2) --++(-9,0) --++(0,-9) --++(2,0) --++(0,2) --++(1,0) --++(0,1) --++(1,0) -- cycle;
\draw[dashed] (-4,0) -- (0,0) -- (0,4);
\draw[blue] (-2,3) rectangle (-1,2);
\node[blue] at (-1.5,2.5) {$x$};
\draw[blue] (-1,3) rectangle (5,2);
\draw[blue] (-2,2) rectangle (-1,-3);
\end{scope}
\end{tikzpicture}
\caption{An alternative description of $\rk(i,j)$}
\label{fig:BSTsymmetric}
\end{figure}
\end{remark}
\end{comment}

\subsection{Root systems and Weyl groups}
Readers are referred to \cite{humphreys} for detailed exposition on root systems and Weyl groups. Let $\Phi\subset V\simeq\mathbb{R}^d$ be a finite crystallographic root system of rank $d$, with a chosen set of positive roots $\Phi^+$ which corresponds to a set of simple roots $\Delta=\{\alpha_0,\alpha_1,\ldots,\alpha_{d-1}\}$. Let $s_{\alpha}$ be the reflection across the hyperplane normal to $\alpha$, and write $s_i$ for the simple reflections $s_{\alpha_i}$. Let $W(\Phi)\subset\mathrm{GL}(V)$ be the finite Weyl group, defined to be generated by $s_0,\ldots,s_{d-1}$. 

For $w\in W(\Phi)$, let $\ell(w)$ denote its Coxeter length, which equals the size of its (left) inversion set $\Inv(w):=\Phi^+\cap w\Phi^-$. For any sequence $\mathbf{a} = (a_1,a_2,\ldots,a_{\ell(w)})$, we say $\mathbf{a}$ is a reduced word of $w$ if $w = s_{a_1}s_{a_2},\ldots,s_{a_{\ell(w)}}$. 
Let $\Red(w)$ be the set of reduced words of $w$. For each reduced word $\mathbf{a}\in \Red(w)$, its \textit{(total) reflection order} is an ordering $\ro(\mathbf{a})=\gamma_1,\ldots,\gamma_{\ell(w)}$ of $\Inv(w)$ where $\gamma_j=s_{a_1}\cdots s_{a_{j-1}}\alpha_j\in\Phi^+$. 
Let 
\[\ro(w) = \{\ro(\mathbf{a}):\mathbf{a}\in \Red(w)\}.\]
The following proposition is classical and very useful, which follows immediately from the biconvexity classification of inversion sets. See for example Proposition 3 of \cite{bjorner1984orderings}.
\begin{prop}\label{prop:root-ordering}
Let $\gamma = \gamma_1,\ldots,\gamma_{\ell(w)}$ be an ordering of $\Inv(w)$. Then $\gamma\in \ro(w)$ if and only if for all the triples $\alpha,\beta,\alpha+\beta\in\Phi^+$ such that $\alpha,\alpha+\beta\in\Inv(w)$,
\begin{enumerate}
\item if $\beta\notin\Inv(w)$, then $\alpha$ appears before $\alpha+\beta$ in this sequence;
\item and if $\beta\in\Inv(w)$, then $\alpha+\beta$ appears in the middle of $\alpha$ and $\beta$. 
\end{enumerate}
\end{prop}

We are primarily concerned with root systems of type $B_n$, and adopt the following convention, where $e_i$ is the $i$-th coordinate vector:
\begin{itemize}
\item $\Phi(B_n)=\{\pm e_j\pm e_i\:|\: 1\leq i<j\leq n\}\cup\{\pm e_i\:|\: 1\leq i\leq n\}$;
\item $\Phi^+(B_n)=\{e_j\pm e_i\:|\: 1\leq i<j\leq n\}\cup\{e_i\:|\: 1\leq i\leq n\}$;
\item $\Delta=\{\alpha_0=e_1,\alpha_1=e_2-e_1,\ldots,\alpha_{n-1}=e_{n}-e_{n-1}\}$;
\item $W(B_n)=\{\text{permutation }w\text{ on }1,\ldots,n,\bar 1,\ldots,\bar n\:|\: w(i)=-w(\bar i),\ \forall i\}.$
\end{itemize}
The type $B_n$ Weyl group $W(B_n)$ is called the group of \textit{signed permutations}. For a signed permutation $w$, its one-line notation is written as $w(1)w(2)\cdots w(n)$. For example, $w=3\bar 42\bar 1\in W(B_4)$ means that $w(1)=3$, $w(2)=-4$, $w(3)=2$ so that $w(-3)=-2$ and $w(4)=-1$. A reduced word of $w\in W(B_n)$ can be viewed as going from $\mathrm{id}=12\cdots n$ to $w$ by swapping adjacent entries (and their negatives) one step at a time, while the corresponding reflection order records $e_j-e_i$ if the values $j$ and $i$ are swapped (and records $e_i$ if $i$ and $\bar i$ are swapped).
\begin{ex}
Consider $w=1\bar342\in W(B_4)$ with a reduced word 
$\mathbf{a} = 21031\in \Red(w)$.
%$R=s_2s_1s_0s_3s_1\in\Red(w)$
We compute its reflection order to be $e_3-e_2,e_3-e_1,e_3,e_4-e_2,e_3+e_1$, which can be seen as follows:
\[
\begin{tikzcd}
1234 \arrow[r, "e_3-e_2"]& 1324 \arrow[r, "e_3-e_1"] & 3124 \arrow[r, "e_3"] & \bar3124 \arrow[r, "e_4-e_2"] & \bar3142 \arrow[r, "e_3+e_1"] & 1\bar342.
\end{tikzcd}
\]
\end{ex}