A bijection between dominant Shi regions and core partitions

Fishel, Susanna; Vazirani, Monica

doi:10.1016/j.ejc.2010.05.014

Cited by 26 publications

(30 citation statements)

References 18 publications

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“…This question has been answered affirmatively by Vandehey in an unpublished thesis [10]; in the present paper we give a simpler proof. Another avenue is pursued by Fishel and Vazirani [3], who examine (s, t)-cores in connection with alcove geometry in the cases where t ≡ ±1 (mod s), exhibiting natural bijections between (s, t)-cores and (bounded) regions in the extended Shi arrangement.…”

Section: Introductionmentioning

confidence: 99%

The t-core of an s-core

Fayers

2011

Journal of Combinatorial Theory, Series A

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We consider the $t$-core of an $s$-core partition, when $s$ and $t$ are coprime positive integers. Olsson has shown that the $t$-core of an $s$-core is again an $s$-core, and we examine certain actions of the affine symmetric group on $s$-cores which preserve the $t$-core of an $s$-core. Along the way, we give a new proof of Olsson's result. We also give a new proof of a result of Vandehey, showing that there is a simultaneous $s$- and $t$-core which contains all others

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Section: Introductionmentioning

confidence: 99%

The t-core of an s-core

Fayers

2011

Journal of Combinatorial Theory, Series A

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“…However, his discovery that the inverses of the minimal permutations correspond to a simplex is worth the price of admission. The minimal alcoves have been useful in other enumeration; see [Ath05,FV10], for example. For another example, Hohlweg, Nadeau, and Williams, in [HNW16], generalize the Shi arrangement to any Coxeter group (and beyond!)…”

Section: Enumerationmentioning

confidence: 99%

“…We mention that Fishel and Vazirani mapped partitions which are both n and nm + 1 cores to dominant regions in the m-Shi arrangment of type A n−1 in [FV10] using abacus diagrams and the root lattice. ∅ FIGURE 4.6.…”

Section: Enumerationmentioning

confidence: 99%

A Survey of the Shi Arrangement

Fishel

2019

Association for Women in Mathematics Series

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In [Shi86], Shi proved Lusztig's conjecture that the number of two-sided cells for the affine Weyl group of type A n−1 is the number of partitions of n. As a byproduct, he introduced the Shi arrangement of hyperplanes and found a few of its remarkable properties. The Shi arrangement has since become a central object in algebraic combinatorics. This article is intended to be a fairly gentle introduction to the Shi arrangement, intended for readers with a modest background in combinatorics, algebra, and Euclidean geometry. After background material in Section 1, this introduction to the arrangement will be by way of a discussion in Section 2 of how it arose, some of its marvelous enumerative properties in Section 3, and some of its surprising connections to algebra in Section 4. For some brief comments on recent extensions, see Section 5 and for an incomplete list of topics we left out, see Section 6.

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“…Simultaneous cores have numerous applications in algebraic combinatorics. For instance, Susanna Fishel and Monica Vazirani [FV09,FV10] showed that when t = ds ± 1 for some d ∈ N, they are naturally in bijection with certain regions of the d-Shi arrangement in type A. Drew Armstrong, Christopher Hanusa, and Brant Jones [AHJ14] extended this work to type C and related simultaneous cores to rational Catalan combinatorics.…”

Section: Introductionmentioning

confidence: 99%

The Asymptotic Normality of $(s,s+1)$-Cores with Distinct Parts

Komlós

Sergel

Tusnády

2020

Electron. J. Combin.

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Simultaneous core partitions are important objects in algebraic combinatorics. Recently there has been interest in studying the distribution of sizes among all (s, t)-cores for coprime s and t. Zaleski (2017) gave strong evidence that when we restrict our attention to (s, s + 1)cores with distinct parts, the resulting distribution is approximately normal. We prove his conjecture by applying the Combinatorial Central Limit Theorem and mixing the resulting normal distributions.

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A bijection between dominant Shi regions and core partitions

Cited by 26 publications

References 18 publications

The t-core of an s-core

The t-core of an s-core

A Survey of the Shi Arrangement

The Asymptotic Normality of $(s,s+1)$-Cores with Distinct Parts

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