Partition Statistics Equidistributed with the Number of Hook Difference One Cells

TL;DR

This paper refines a known equidistribution result relating hook difference statistics in partitions to the largest repeated part, introduces combinatorial proofs, and proposes new q,t-Catalan numbers with combinatorial interpretations.

Contribution

It provides a combinatorial refinement of Buryak and Feigin's theorem, offers new formulas for q-Catalan numbers, and introduces a novel q,t-Catalan number with a clear combinatorial meaning.

Findings

Hook difference statistic $h_{1,1}$ is equidistributed with the largest repeated part $a_2$.

A new combinatorial formula for q-Catalan numbers is derived.

Introduction of a new q,t-Catalan number with a simple combinatorial interpretation.

Abstract

Let $\lambda$ be a partition, viewed as a Young diagram. We define the hook difference of a cell of $\lambda$ to be the difference of its leg and arm lengths. Define $h_{1,1}(\lambda)$ to be the number of cells of $\lambda$ with hook difference one. In the paper of Buryak and Feigin (arXiv:1206.5640), algebraic geometry is used to prove a generating function identity which implies that $h_{1,1}$ is equidistributed with $a_2$, the largest part of a partition that appears at least twice, over the partitions of a given size. In this paper, we propose a refinement of the theorem of Buryak and Feigin and prove some partial results using combinatorial methods. We also obtain a new formula for the q-Catalan numbers which naturally leads us to define a new q,t-Catalan number with a simple combinatorial interpretation.