On rotated Schur-positive sets

Elizalde, Sergi; Roichman, Yuval

doi:10.1016/j.jcta.2017.06.002

Cited by 13 publications

(19 citation statements)

References 20 publications

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“…An analogous phenomenon for certain disconnected shapes has recently been discovered in [9]. For a partition λ of n − 1, let λ be the skew shape obtained from the Young diagram of shape λ by placing a disconnected cell at its upper right corner.…”

Section: Introductionmentioning

confidence: 60%

“…A variant of promotion which shifts the cyclic descent set for hooks plus a disconnected cell was described in [9], see Section 2.2. In this section we present a bijection on SYT of a shape consisting of a hook plus an internal cell, cyclically shifting their (uniquely defined) cyclic descent sets.…”

Section: The Bijection φ For Near-hooksmentioning

confidence: 99%

“…For some shapes, namely rectangles [19], straight shapes with a disconnected box added in the upper-right corner [9], strips, hooks plus one cell, and two-row straight shapes, we have been able to explicitly describe a bijection φ : SYT(λ/µ) → SYT(λ/µ) with the property that cDes(φT ) = 1 + cDes(T ) for every T ∈ SYT(λ/µ). In general, finding such a bijection is not immediate, even after a combinatorial definition of cDes has been found; for example, we have no explicit φ for two-row skew shapes.…”

Section: Shapementioning

confidence: 99%

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Cyclic descents for near-hook and two-row shapes

Adin

Elizalde

Roichman

2019

European Journal of Combinatorics

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A notion of cyclic descents on standard Young tableaux (SYT) of rectangular shape was introduced by Rhoades, and extended to certain skew shapes by the last two authors. The cyclic descent set restricts to the usual descent set when the largest value is ignored, and has the property that the number of SYT of a given shape with a given cyclic descent set D is invariant under cyclic shifts of the entries of D. Following these results, the existence of cyclic descent sets for standard Young tableaux of any skew shape other than a ribbon was conjectured by the authors, and recently proved by Adin, Reiner and Roichman. Unfortunately, the proof does not provide a natural definition of the cyclic descent set for a specific tableau.In this paper we explicitly describe cyclic descent sets and resulting generating functions for SYT of (possibly skew) shapes which either have exactly two rows or are near-hooks, i.e., are one cell away from a hook. Our definition provides a constructive combinatorial proof of the existence of cyclic descent sets for these shapes, and coincides with that of Rhoades for two-row rectangular shapes. We also show that cyclic descent sets for near-hook shaped tableaux are unique.

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Section: Introductionmentioning

confidence: 60%

Section: The Bijection φ For Near-hooksmentioning

confidence: 99%

Section: Shapementioning

confidence: 99%

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Cyclic descents for near-hook and two-row shapes

Adin

Elizalde

Roichman

2019

European Journal of Combinatorics

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“…Theorem 6.18. [5,Proposition 5.3] Let λ/µ consist of a Young diagram of size n − 1 and a disconnected northeast cell. For T ∈ SYT(λ/µ), let n ∈ cDes(T ) if and only if one of the following holds.…”

Section: Recall Thatmentioning

confidence: 99%

“…[2,13,15] define promotion to be the operation that we term demotion. We follow the definition from [5,12].…”

Section: Introductionmentioning

confidence: 99%

Cyclic descents for general skew tableaux

Huang

2020

Journal of Combinatorial Theory, Series A

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A cyclic descent function on standard Young tableaux of size n is a function that restricts to the usual descent function when n is omitted, such that the number of standard Young tableaux of given shape with cyclic descent set D ⊂ [n] is invariant under any modulo n shift of D. The notion of cyclic descent was first studied for rectangles by Rhoades, and then generalized to certain families of skew shapes by Adin, Elizalde, and Roichman. Adin, Reiner, and Roichman proved that a skew shape has a cyclic descent map if and only if it is not a connected ribbon. Unfortunately, their proof is nonconstructive; until now, explicit cyclic descent maps are known only for small families of shapes.In this paper, we construct an explicit cyclic descent map for all shapes where this is possible. We thus provide a constructive proof of Adin, Reiner, and Roichman's result. Our construction of a cyclic descent map generalizes many of the constructions in the literature.

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Schur-positive sets of permutations via products and grid classes

Elizalde

Roichman²

2016

J Algebr Comb

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Characterizing sets of permutations whose associated quasisymmetric function is symmetric and Schur-positive is a long-standing problem in algebraic combinatorics. In this paper we present a general method to construct Schur-positive sets and multisets, based on geometric grid classes and the product operation. Our approach produces many new instances of Schurpositive sets, and provides a broad framework that explains the existence of known such sets that until now were sporadic cases.

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On rotated Schur-positive sets

Cited by 13 publications

References 20 publications

Cyclic descents for near-hook and two-row shapes

Cyclic descents for near-hook and two-row shapes

Cyclic descents for general skew tableaux

Schur-positive sets of permutations via products and grid classes

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