Average Size of a Self-conjugate (s, t)-Core Partition

Chen, William Y. C.; Huang, Harry H. Y.; Wang, Larry X. W.

doi:10.48550/arxiv.1405.2175

Cited by 4 publications

(6 citation statements)

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“…Stanley and Zanello [11] showed that the average size of an (s, s + 1)-core equals s+1 3 /2. Chen, Huang and Wang [5] proved the conjecture for the average size of a self-conjugate (s, t)-core.…”

Section: Introductionmentioning

confidence: 90%

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

Yang

Zhong

Zhou

2015

European Journal of Combinatorics

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Anderson established a connection between core partitions and order ideals of certain posets by mapping a partition to its β-set. In this paper, we give a characterization of the poset P (s,s+1,s+2) whose order ideals correspond to (s, s + 1, s + 2)-core partitions. Using this characterization, we obtain the number of (s, s + 1, s + 2)-core partitions, the maximum size and the average size of an (s, s + 1, s + 2)-core partition, confirming three conjectures posed by Amdeberhan.

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Section: Introductionmentioning

confidence: 90%

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

Yang

Zhong

Zhou

2015

European Journal of Combinatorics

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“…Johnson and Chen, Huang, and Wang give proofs that the self-conjugate (a, b)-core partitions have the same average size as the set of all (a, b)-core partitions [11,19]. It is natural to ask whether sizes of other subfamilies of (a, b)-cores have similar statistical properties.…”

Section: Bothmentioning

confidence: 99%

“…A partition is self-conjugate if it is equal to its conjugate. Armstrong's conjecture was proven for self-conjugate (a, b)-cores by Chen, Huang, and Wang [11]. After these partial results, the full theorem was proven by Johnson [19].…”

Section: Introductionmentioning

confidence: 97%

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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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“…A simpler proof was provided by Tripathi [17]. More results on (s, t)-core partitions can be found in [2,4,6,10,16,18,19].…”

Section: Introductionmentioning

confidence: 99%

The Raney numbers and(s,s+1)-core partitions

Zhou

Yan

2017

European Journal of Combinatorics

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The Raney numbers R p,r (k) are a two-parameter generalization of the Catalan numbers. In this paper, we obtain a recurrence relation for the Raney numbers which is a generalization of the recurrence relation for the Catalan numbers. Using this recurrence relation, we confirm a conjecture posed by Amdeberhan concerning the enumeration of (s, s+1)-core partitions λ with parts that are multiples of p. We then give a new combinatorial interpretation for the Raney numbers R p+1,r+1 (k) with 0 ≤ r < p in terms of (kp + r, kp + r + 1)-core partitions λ with parts that are multiples of p.

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Average Size of a Self-conjugate (s, t)-Core Partition

Cited by 4 publications

References 0 publications

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

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

Numerical Sets, Core Partitions, and Integer Points in Polytopes

The Raney numbers and(s,s+1)-core partitions

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