[Paper Review] Increasing tableaux and Narayana numbers

Timothy Pressey, Anna Stokke+1 more|arXiv (Cornell University)|Sep 9, 2014

Advanced Combinatorial Mathematics5 references3 citations

TL;DR

This paper establishes a connection between increasing rectangular tableaux and generalized Narayana numbers, proving that the number of such tableaux of shape $m \times n$ equals the small $m$-Schröder number, which is the evaluation of the $m$-Narayana polynomial at $t=2$. It further introduces a $q$-analogue of the counting formula and demonstrates a cyclic sieving phenomenon for $K$-promotion on increasing hook tableaux using a combinatorial interpretation of arm–leg inversions.

ABSTRACT

We give a counting formula for the set of rectangular increasing tableaux in terms of generalized Narayana numbers. We define small $m$-Schröder paths and give a bijection between the set of increasing rectangular tableaux and small $m$-Schröder paths, generalizing a result of Pechenik [3]. Using $K$-jeu de taquin promotion, which was defined by Thomas and Yong [10], we give a cyclic sieving phenomenon for the set of increasing hook tableaux.

Motivation & Objective

• To generalize Pechenik's bijection between increasing tableaux of shape $2 \times n$ and small Schröder paths to $m \times n$ rectangular shapes.
• To derive a formula for the number of increasing tableaux of shape $m \times n$ in terms of generalized Narayana numbers.
• To establish a cyclic sieving phenomenon (CSP) for $K$-jeu de taquin promotion on increasing hook tableaux using a $q$-analogue of the counting formula.

Proposed method

• Define small $m$-Schröder paths in $m$-dimensional space as lattice paths from $(0,\dots,0)$ to $(n,\dots,n)$ with steps $X_i$, constrained by $x_m \leq \cdots \leq x_1$, and with $\ell$ ascents.
• Construct a bijection between small $m$-Schröder paths and increasing rectangular tableaux of shape $m \times n$, generalizing Pechenik's result for $m=2$.
• Use $K$-jeu de taquin promotion to define a cyclic group action on increasing hook tableaux of shape $(N-r,1^r)$, with group order $N-k-1$.
• Define a $q$-analogue polynomial $X(q) = \left[\begin{smallmatrix}N-k-1 \\ r\end{smallmatrix}\right]_q \left[\begin{smallmatrix}r \\ k\end{smallmatrix}\right]_q$ that counts tableaux by arm–leg inversions.
• Prove that the triple $(\mathop{\rm Inc}\nolimits_k(N-r,1^r), C, X(q))$ exhibits the cyclic sieving phenomenon, where $C$ is the cyclic group generated by $K$-promotion.
• Use a map $\psi$ from increasing hook tableaux to standard hook tableaux and leverage results from Reiner, Stanton, and White to verify the CSP.

Experimental results

Research questions

• RQ1 How can the number of increasing rectangular tableaux of shape $m \times n$ be expressed in terms of generalized Narayana numbers?
• RQ2 Is there a generalization of Pechenik's bijection between increasing tableaux and small Schröder paths to higher dimensions ($m > 2$)?
• RQ3 Does a $q$-analogue of the counting formula for increasing tableaux exhibit the cyclic sieving phenomenon under $K$-promotion?
• RQ4 What is the combinatorial interpretation of the coefficients in the $q$-analogue polynomial used in the CSP?
• RQ5 Why does the natural $q$-analogue of the formula for $|\mathop{\rm Inc}\nolimits_1(3 \times 3)|$ fail to satisfy the CSP under $K$-promotion?

Key findings

• The number of increasing tableaux of shape $m \times n$ is equal to the small $m$-Schröder number $N_{m,n}(2)$, generalizing Pechenik's result for $m=2$.
• The cardinality of $\mathop{\rm Inc}\nolimits_k(m \times n)$ is a linear combination of generalized Narayana numbers $N(m,n,\ell)$, with a specific formula given in Theorem 2.4.
• A key corollary states that $|\mathop{\rm Inc}\nolimits_1(m \times n)| = \frac{(m-1)(n-1)}{2} |\mathop{\rm SYT}\nolimits(m \times n)|$, linking increasing tableaux with standard tableaux.
• The $q$-analogue polynomial $X(q) = \left[\begin{smallmatrix}N-k-1 \\ r\end{smallmatrix}\right]_q \left[\begin{smallmatrix}r \\ k\end{smallmatrix}\right]_q$ counts increasing hook tableaux by arm–leg inversions and satisfies the cyclic sieving phenomenon under $K$-promotion.
• The coefficients of the $q$-analogue polynomial have a natural combinatorial interpretation as the number of arm–leg inversions in increasing hook tableaux.
• An example shows that the natural $q$-analogue of the formula for $|\mathop{\rm Inc}\nolimits_1(3 \times 3)|$ does not yield an integer value at a primitive 8th root of unity, thus failing to satisfy the CSP, highlighting the non-triviality of the $q$-analogue choice.