Schubert polynomials, 132-patterns, and Stanley’s conjecture

Weigandt, Anna

doi:10.5802/alco.27

Cited by 8 publications

(10 citation statements)

References 4 publications

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“…, 1) which has been extensively studied in recent years (see e.g. [BHY,MeS,SeS,Sta4,Wei,Woo]). Below we present two such formulas: Here σ × ω and σ ⊗ ω are the direct sum and the Kronecker product of permutations σ and ω (see §2.2).…”

Section: Introductionmentioning

confidence: 99%

Hook formulas for skew shapes III. Multivariate and product formulas

Morales¹,

Pak²,

Panova³

2019

Algebraic Combinatorics

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We give new product formulas for the number of standard Young tableaux of certain skew shapes and for the principal evaluation of the certain Schubert polynomials. These are proved by utilizing symmetries for evaluations of factorial Schur functions, extensively studied in the first two papers in the series [MPP1, MPP2]. We also apply our technology to obtain determinantal and product formulas for the partition function of certain weighted lozenge tilings, and give various probabilistic and asymptotic applications.

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Section: Introductionmentioning

confidence: 99%

Hook formulas for skew shapes III. Multivariate and product formulas

Morales¹,

Pak²,

Panova³

2019

Algebraic Combinatorics

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“…Finally, under Theorem 1.3, each F i corresponds to a distinct exponent vector since the sum of the labels is strictly decreasing at each step F i−1 → F i . From Corollary 4.5, this result of A. Weigandt [17] is immediate: As shown in [17], Corollary 4.6 in turn implies S w (1, 1, . .…”

Section: Define a Collection Of Intervals In [N] Bymentioning

confidence: 83%

An efficient algorithm for deciding vanishing of Schubert polynomial coefficients

Adve¹,

Robichaux

Yong

2021

Preprint

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Schubert polynomials form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. The vanishing problem for Schubert polynomials asks if a coefficient of a Schubert polynomial is zero. We give a tableau criterion to solve this problem, from which we deduce the first polynomial time algorithm. These results are obtained from new characterizations of the Schubitope, a generalization of the permutahedron defined for any subset of the n × n grid. In contrast, we show that computing these coefficients explicitly is #P-complete.

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“…A deeper study of Macdonald's reduced word identity and its generalizations has seen renewed interest recently and has brought forth various interesting aspects of the interplay between Schubert polynomials, combinatorics of reduced words, and differential operators on polynomials. We refer the reader to [9,26,62,47] for more details. As we shall see in the next section, an expression rather reminiscent of the right hand side of (2.15) plays a key role in our quest to obtain the Schubert expansion for τ n = [Perm n ], and its appearance in this context begs for deeper explanation.…”

Section: Reduced Wordsmentioning

confidence: 99%

“…Proof of Proposition 6.4. We use here [62,Lemma 3.6(iii)] which states that for any w ∈ S ∞ , a(γ w ) = a(γ w −1 ), which translates into ā(w) = ā(w −1 ). We then conclude by Proposition A.2.…”

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confidence: 99%

The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials

Nadeau¹,

Tewari²

2020

Preprint

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We compute the expansion of the cohomology class of the permutahedral variety in the basis of Schubert classes. The resulting structure constants aw are expressed as a sum of normalized mixed Eulerian numbers indexed naturally by reduced words of w. The description implies that the aw are positive for all permutations w ∈ Sn of length n − 1, thereby answering a question of Harada, Horiguchi, Masuda and Park. We use the same expression to establish the invariance of aw under taking inverses and conjugation by the longest word, and subsequently establish an intriguing cyclic sum rule for the numbers.We then move toward a deeper combinatorial understanding for the aw by exploiting in addition the relation to Postnikov's divided symmetrization. Finally, we are able to give a combinatorial interpretation for aw when w is vexillary, in terms of certain tableau descents. It is based in part on a relation between the numbers aw and principal specializations of Schubert polynomials.Along the way, we prove results and raise questions of independent interest about the combinatorics of permutations, Schubert polynomials and related objects.

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Schubert polynomials, 132-patterns, and Stanley’s conjecture

Cited by 8 publications

References 4 publications

Hook formulas for skew shapes III. Multivariate and product formulas

Hook formulas for skew shapes III. Multivariate and product formulas

An efficient algorithm for deciding vanishing of Schubert polynomial coefficients

The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials

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