Simultaneous Core Partitions: Parameterizations and Sums

Abstract

Fix coprime s, t ≥ 1. We re-prove, without Ehrhart reciprocity, a conjecture of Armstrong (recently verified by Johnson) that the finitely many simultaneous (s, t)-cores have average size 1 24 (s−1)(t−1)(s+t+1), and that the subset of self-conjugate cores has the same average (first shown by Chen-Huang-Wang). We similarly prove a recent conjecture of Fayers that the average weighted by an inverse stabilizer-giving the "expected size of the t-core of a random s-core"-is 1 24 (s − 1)(t 2 − 1). We also prove Fayers' conjecture that the analogous self-conjugate average is the same if t is odd, but instead 1 24 (s − 1)(t 2 + 2) if t is even. In principle, our explicit methods-or implicit variants thereof-extend to averages of arbitrary powers.The main new observation is that the stabilizers appearing in Fayers' conjectures have simple formulas in Johnson's z-coordinates parameterization of (s, t)-cores.We also observe that the z-coordinates extend to parameterize general t-cores. As an example application with t := s + d, we count the number of (s, s + d, s + 2d)-cores for coprime s, d ≥ 1, verifying a recent conjecture of Amdeberhan and Leven.Date: January 12, 2016. v4: added reference [10] and updated [11]; shorter version available at Electronic J. Combin. 23(1) (2016), #P1.4. v3: minor improvements and clarifications throughout, including implicit variant details; added some references and remarks on finite beta-sets, maximal cores, and poset method. v2: extended z-coordinates of Johnson to parameterize all t-cores, not just (s, t)-cores; applied this to a conjecture of Amdeberhan-Leven (v3 mentions later independent proof by Johnson); added some references. Version for the arXiv, with 34 pages, 3 figures. Shorter version with 28 pages (currently available at https://www.overleaf.com/read/ xmzxdgbdcpnq), with fewer details, to be submitted for publication. Comments welcome on either. 1 arXiv:1507.04290v4 [math.CO]

Johnson’s bijections and their application to counting simultaneous core partitions

“…Letting a = s + 1, b 0 = s, b 1 = s + 2, Corollary 5.5 recovers a theorem of Yang-Zhong-Zhou [17]. Letting a = s + d, b 0 = s, b 1 = s + 2d, Corollary 5.5 recovers Theorem 1.6 of Wang [13].…”

“…, b n } contains at least one pair of relatively prime numbers. As a corollary, we obtain an alternative proof for the number of (s, s + d, s + 2d)-core partitions, which was given by Yang-Zhong-Zhou [17] and Wang [13]. Subsequently, we also give a formula for the number of (s, s + d, s + 2d, s + 3d)-core partitions.…”

“…arXiv:1711.01469v1 [math.CO] 4 Nov 2017 2 JINEON BAEK, HAYAN NAM, AND MYUNGJUN YUArmstrong's conjecture by using Ehrhart theory. A proof without Ehrhart theory was given by Wang [13].In [8], Johnson estabilished a bijection between the set of (a, b)-cores and the setBy showing that the cardinality of this set is Cat a,b , he gave a new proof of Anderson's theorem. Inspired by Johnson's method and this bijection, we count the number of simultaneous core partitions.…”

On the polynomiality and asymptotics of moments of sizes for random (n, dn ± 1)-core partitions with distinct parts

“…For example, when t 1 and t 2 are coprime to each other, it was proved by Olsson and Stanton [16] that the largest size of (t 1 , t 2 )-core partitions equals (t 2 1 − 1)(t 2 2 − 1)/24, in their study of block inclusions of symmetric groups. Armstrong (see [4]) gave the following conjecture on the average size of such partitions, which was first proved by Johnson [10] and later by Wang [21]. Theorem 1.2 (Armstrong's Conjecture).…”

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 !…”