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Simple Generating Functions for Certain Young Tableaux with Periodic Walls

Feihu Liu, Guoce Xin

Abstract

Recently, Banderier et. al. considered Young tableaux with walls, which are similar to standard Young tableaux, except that local decreases are allowed at some walls. We count the numbers $\overline{f}_m(n)$ of Young tableaux of shape $2\times mn$ with walls, that allow local decreases at the $(jm+i)$-th columns for all $j=0,\dots, n-1$ and $i=2,\dots, m$. We find that they have nice generating functions (thanks to the OEIS) as follows. $$\overline{F}_m(x)=\sum_{n\geq 0}\overline{f}_m(n)x^n=\prod_{k=1}^{m}C(e^{k\frac{2πi}{m}} x^\frac{1}{m})=\exp \left(\sum_{n\geq 1}\binom{2mn-1}{mn-1}\frac{x^n}{n}\right),$$ where $C(x)=\frac{1-\sqrt{1-4x}}{2x}$ is the well-known Catalan generating function. We prove generalizations of this result. Firstly, we use the Yamanouchi word to transform Young tableaux with horizontal walls into lattice paths. This results in a determinant formula. Then by lattice path counting theory, we obtain the generating functions $F_r(x)$ for the number of lattice paths from $(0,0)$ to $(\ell n-r,kn)$ that never go above the path $(N^kE^{\ell})^{n-1}N^kE^{\ell-r}$, where $N,E$ stand for north and east steps, respectively. We also obtain exponential formulas for $F_1(x)$ and $F_\ell(x)$. The formula for $\overline{F}_m(x)$ is thus proved since it is just $F_1(x)$ specializes at $k=\ell=m$.

Simple Generating Functions for Certain Young Tableaux with Periodic Walls

Abstract

Recently, Banderier et. al. considered Young tableaux with walls, which are similar to standard Young tableaux, except that local decreases are allowed at some walls. We count the numbers of Young tableaux of shape with walls, that allow local decreases at the -th columns for all and . We find that they have nice generating functions (thanks to the OEIS) as follows. where is the well-known Catalan generating function. We prove generalizations of this result. Firstly, we use the Yamanouchi word to transform Young tableaux with horizontal walls into lattice paths. This results in a determinant formula. Then by lattice path counting theory, we obtain the generating functions for the number of lattice paths from to that never go above the path , where stand for north and east steps, respectively. We also obtain exponential formulas for and . The formula for is thus proved since it is just specializes at .
Paper Structure (10 sections, 30 theorems, 119 equations, 3 figures)

This paper contains 10 sections, 30 theorems, 119 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Rectangular Young Tableaux with Horizontal Walls
  3. Lattice Path Enumeration
  4. Symmetric Functions
  5. Lattice Paths
  6. Exponential Formulas for $Q(x)$ and $F_1(x)$
  7. Two Byproducts
  8. Recursive Operation
  9. A Symmetric Function Formula
  10. Concluding Remark

Key Result

Theorem 1.1

Suppose $m$ is a positive integer. Let $\overline{f}_m(n)$ denote the number of Young tableaux with periodic walls over $\mathcal{B}_m^n$, with convention $\overline{f}_m(0)=1$. Then we have where $\xi=e^{\frac{2\pi i}{m}}$ and $C(x)$ is the Catalan generating function.

Figures (3)

  • Figure 1: A Young tableaux of shape $\lambda=(6,6)$ with walls.
  • Figure 2: A Young tableaux with periodic walls of a block $\mathcal{B}_2$ of shape $(2,2)$.
  • Figure 4: An example of bijection for $\mathcal{B}_3^2$ in Proposition \ref{['DeterFormuFM']}.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4: G.Kreweras
  • ...and 37 more