Partition Statistics Equidistributed with the Number of Hook Difference One Cells
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 qCatalan numbers which naturally leads us to define a new q,tCatalan number with a simple combinatorial interpretation.
 Publication:

arXiv eprints
 Pub Date:
 April 2014
 arXiv:
 arXiv:1405.0072
 Bibcode:
 2014arXiv1405.0072H
 Keywords:

 Mathematics  Combinatorics;
 05A15;
 05A17;
 05A19;
 05A30