  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.
