Multi-cores, posets, and lattice paths

Abstract

Hooks are prominent in representation theory (of symmetric groups) and they play a role in number theory (via cranks associated to Ramanujan's congruences). A partition of a positive integer n has a Young diagram representation. To each cell in the diagram there is an associated statistic called hook length, and if a number t is absent from the diagram then the partition is called a t-core. A partition is an (s, t)-core if it is both an s-and a t-core. Since the work of Anderson on (s, t)-cores, the topic has received growing attention. This paper expands the discussion to multiple-cores. More precisely, we explore (s, s + 1, . . . , s + k)-core partitions much in the spirit of a recent paper by Stanley and Zanello. In fact, our results exploit connections between three combinatorial objects: multi-cores, posets and lattice paths (with a novel generalization of Dyck paths). Additional results and conjectures are scattered throughout the paper. For example, one of these statements implies a curious symmetry for twin-coprime (s, s + 2)-core partitions.

Bijections between t-core partitions and t-tuples

“…Amdeberhan [1] used this formula to obtain some identities for Catalan numbers. Besides, Lemma 3.3 leads to many interesting corollaries.…”

“…Core partitions of numerous types of additional restrictions have long been studied, since they are closely related to the representation of symmetric group [15], the theory of cranks [13], Dyck-paths [1,3,28], and Euler's theorem [22]. To solve core problems, mathematicians provide many different tools, including t-abacus [3,15], Hasse diagram [27,28] and even ideas from quantum mechanics [16].…”

“…Proof. (1) Taking (a, b, c 1 , · · · , c n ) = (s + 1, s, s + 2, s + 3, · · · , s + r) into Theorem 3.7, we have d i = r s+1 (i) for 1 ≤ i ≤ r − 1. Then there is a bijection from P s,s+1,...,s+r to Z s+1,s…”

On the largest sizes of certain simultaneous core partitions with distinct parts

“…Therefore it is a (6, 8)-core partition since none of its hook lengths is divisible by 6 or 8. Simultaneous core partitions have been widely studied in the past fifteen years (see [2,4,7,8,9,12,14,15,17,19,20,23]) since Anderson's work [3], who showed that the number of (t 1 , t 2 )-core partitions is equal to (t 1 + t 2 − 1)!/(t 1 ! t 2 !…”

On the number of simultaneous core partitions with d-distinct parts

“…. Other results in this area are due to Amdeberhan and Leven [5], Yang, Zhong, and Zhou [22], Aggarwal [1], and Wang [19].…”