On the enumeration of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.gif" display="inline" overflow="scroll"><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>s</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math>-core partitions

Yang, Jane Y. X.; Zhong, Michael X. X.; Zhou, Robin D. P.

doi:10.1016/j.ejc.2015.03.012

Cited by 38 publications

(38 citation statements)

References 13 publications

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“…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].…”

Section: Counting Simultaneous Core Partitionsmentioning

confidence: 53%

“…, 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.…”

mentioning

confidence: 79%

“…If s is odd, Yang, Zhong, and Zhou [17] showed there are two (s, s + 1, s + 2)-cores (a pair of conjugate partitions) with the largest size, whereas there is a unique self-conjugate (s, s + 1, s + 2)-core with the largest size (see Theorem 3.3).…”

Section: Review Of Johnson's Bijectionsmentioning

confidence: 99%

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

Baek

Nam

2019

European Journal of Combinatorics

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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.

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Section: Counting Simultaneous Core Partitionsmentioning

confidence: 53%

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confidence: 79%

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

Baek

Nam

2019

European Journal of Combinatorics

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“…Authors of [28] gave the recurrence relation above for r = 2. Thus, we have generalized their result.…”

Section: Proof Lemma 32 Provides the Bijection Between (1) And (2)mentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

Bijections between t-core partitions and t-tuples

Zhong

2020

Discrete Mathematics

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This note introduces some bijections relating core partitions and tuples of integers. We apply these bijections to count the number of cores with various types of restriction, including fixed number of parts, limited size of parts, parts divisible by some integer, and distinct parts. For example, we prove that the number of 2t-core partitions into l even parts equals the number of t-core partitions into l parts. We also generalize one expression for simultaneous cores, which was given by Baek, Nam and Yu, recently. Subsequently, we use this expression to obtain recurrence satisfied by numbers of (s, s + 1, · · · , s + r)−core partitions for s ≥ 1.

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Numerical Sets, Core Partitions, and Integer Points in Polytopes

Constantin

Houston-Edwards

Kaplan

2017

Springer Proceedings in Mathematics &Amp; Statistics

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We study a correspondence between numerical sets and integer partitions that leads to a bijection between simultaneous core partitions and the integer points of a certain polytope. We use this correspondence to prove combinatorial results about core partitions. For small values of a, we give formulas for the number of (a, b)-core partitions corresponding to numerical semigroups. We also study the number of partitions with a given hook set.

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On the enumeration of (s,s+1,s+2)-core partitions

Cited by 38 publications

References 13 publications

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

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

Bijections between t-core partitions and t-tuples

Numerical Sets, Core Partitions, and Integer Points in Polytopes

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