Partition Statistics Equidistributed with the Number of Hook Difference One Cells

Abstract

Let λ be a partition, viewed as a Young diagram. We define the hook difference of a cell of λ to be the difference of its leg and arm lengths. Define h 1,1 (λ) to be the number of cells of λ with hook difference one. In [BF], 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 [BF] 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.

On certain unimodal sequences and strict partitions

“…As an immediate application of last theorem, we derive the following result first obtained by Huang et al in [8].…”

“…Corollary 3.7 (Proposition 5.13, [8]). Let P j be the set of all partitions with 2-core size 2j 2 , then we have…”

“…Vandervelde then provided such a bijective proof for the case when the characteristic or BG-rank k = 0, for which partition he named as "balanced partition". The cheif aim of this paper is to prove the entire conjecture bijectively for all integer k. We note that, being not aware of Vandervelde's conjecture, Huang et al [8] supplied essentially the first proof of it (see Corollary 3.7 below).…”

“…(13, 6, 5, 4, 3, 2), (11, 8, 5, 4, 3, 2), (9,8,7,4,3,2). (13, 2, 1), (11, 4, 1), (9, 6, 1), (9,4,3), (7,6,3).…”

“…The authors wish to acknowledge Dennis Stanton for bringing [8] to their attention, and acknowledge Guo-Niu Han and Huan Xiong for their helpful comments on a preliminary version of this paper. Both authors' research were supported by the National Science Foundation of China grant 11501061.…”