The Corners of Core Partitions

Source abstract

In this paper, we are concerned with counting corners of core partitions. We introduce the concepts of stitches and antistitches, which are pairs of cells in a quotient space which we call wrap-up space. We prove that the antistitches of a rational Dyck path are in bijection with the segments of structure sets of the corresponding core partition; therefore the corners of a core partition can be counted by the number of stitches or antistitches. Based on these results, for coprime positive integers $a$ and $b$, we give two essentially different formulae for the number of corners in all $(a,b)$-cores. This leads to an unexpected identity, expressing the rational Catalan numbers as weighed sums of binomial numbers. Moreover, we show that for an $(n,n+1)$-core partition $\lambda$ determined by a certain $(n,n+1)$-Dyck path $P$, the corners of $\lambda$ correspond to pairs of consecutive right steps in $P$. As a consequence, we show that the number of $(n,n+1)$-cores with $k$ corners is the Narayana number $N(n,k+1)$. We also extend these results to multicores.