The Corners of Core Partitions

Abstract

In this paper, we are concerned with counting corners of core partitions. We introduce the concepts of stitches and anti-stitches, which are pairs of cells in a quotient space which we call wrap-up space. We prove that the anti-stitches 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 anti-stitches. 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 λ determined by a certain (n, n + 1)-Dyck path P , the corners of λ 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 multi-cores.

Combinatorics on Bounded Free Motzkin Paths and its Applications

“…, t p . Huang and Wang [11] enumerated the number of (t, t + 1)-cores, (t, t + 1, t + 2)-cores with the fixed number of corners, where these results are generalized to (t, t + 1, • • • , t + p)-cores in [4]. As far as we know, it seems new to get the formula for the number of t-core partitions with the fixed number of corners, which we enumerate this by using the path interpretation.…”

Proof of a Conjecture of Nath and Sellers on Simultaneous Core Partitions

“…Also, simultaneous core partitions are connected with Motzkin paths and Dyck paths (see [9,11,39,40]). Some statistics of simultaneous core partitions, such as numbers of partitions, numbers of corners, largest sizes and average sizes, have attracted much attention in the past twenty years (see [2,3,4,7,8,10,13,14,15,18,20,21,23,24,25,29,31,33,35,37,42,43,44]). For example, Anderson [3] showed that the number of (s 1 , s 2 )-core partitions is equal to 1 s 1 +s 2 s 1 +s 2 s 1…”