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

Partition Statistics Equidistributed with the Number of Hook Difference One Cells