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

Abstract

Johnson recently proved Armstrong's conjecture which states that the average size of an (a, b)-core partition is (a+b+1)(a−1)(b−1)/24. He used various coordinate changes and one-to-one correspondences that are useful for counting problems about simultaneous core partitions. We give an expression for the number of (b 1 , b 2 , · · · , bn)-core partitions where {b 1 , b 2 , · · · , bn} contains at least one pair of relatively prime numbers. We also evaluate the largest size of a self-conjugate (s, s + 1, s + 2)-core partition. 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. We find a general expression for the number of (b 1 , b 2 , . . . , b n )-core partitions where {b 1 , b 2 , . . . , 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. Many authors have studied core partitions satisfying additional restrictions. For example, Berg and Vazirani [7] gave a formula for the number of a-core partitions with largest part x. We generalize this formula, giving a formula for the number of a-core partitions with largest part x and second largest part y.This paper also includes a result related to the largest size of a simultaneous core partition which has been studied by many mathematicians. For example, Aukerman, Kane and Sze [6, Conjecture 8.1] conjectured that if a and b are coprime, the largest size of an (a, b)-core partition is (a 2 − 1)(b 2 − 1)/24. This was proved by Tripathi in [12]. It is natural to wonder what would be the largest size of an (a, b, c)-core. Yang-Zhong-Zhou [17] found a formula for the largest size of an (s, s + 1, s + 2)-core. In section 4, we give a formula for the largest size of a selfconjugate (s, s + 1, s + 2)-core partition. We also prove that such a partition is unique (see Theorem 3.3).The layout of this paper is as follows. In Section 2, we introduce Johnson's ccoordinates and x-coordinates for core partitions. In Section 3, we give a formula for the largest size of a self-conjugate (s, s + 1, s + 2) core partition. In Section 4, using c-coordinates, we count the number of a-core partitions with given largest part and second largest part. In Section 5, we derive formulas for the number of simultaneous core partitions by using Johnson's z-coordinates.

Bijections between t-core partitions and t-tuples

“…We say a partition is the conjugate of λ, denoted by Hence we obtain the following corollary which was proved by Baek, Nam, and Yu [7].…”

“…As a consequence, we surprisingly find that the number of 2t-core partitions into l even parts equals the number of t-core partitions into l parts. In Section 3, inspired by a recent expression for simultaneous core partitions ( [7]), we give a new representation for simultaneous cores, which is useful in studying the topics on (s, s + 1, · · · , s + r)-cores. In the last section, we conclude this note with some formulas for the core partitions with distinct parts.…”

Self-conjugate $(s,s+d,\dots,s+pd)$-core partitions and free rational Motzkin paths

“…In particular, the number of (s, s + 1)-core partitions is the sth Catalan number C s = 1 s+1 2s s = 1 2s+1 2s+1 s . Going further, researchers considered simultaneous core partitions when the sequence of s i forms an arithmetic progression (see [1,4,6,16,17,18]). Recently, the authors [7] enumerated the (s, s + d, .…”

The $(s,s+d,\dots,s+pd)$-core partitions and the rational Motzkin paths

“…Since the work of Anderson, results on (s, t)-cores were published by many researchers (see [3,6,8,9,10,11]). Also, some researchers concerned with simultaneous core partitions whose cores line up in arithmetic progression (see [1,4,7,19,20,21]). Yang-Zhong-Zhou [21] showed that the number of (s, s + 1, s + 2)-core partitions is the sth Motzkin number M s = ⌊s/2⌋ k=0 s 2k C k , where C k is the kth Catalan number.…”

“…Recently, Baek-Nam-Yu [4] obtained an alternative proof for Theorem 1.3 and found a formula for the number of (s, s + d, s + 2d, s + 3d)-core partitions.…”