Excedances in classical and affine Weyl groups

Mongelli, Pietro

doi:10.1016/j.jcta.2013.03.006

Cited by 12 publications

(11 citation statements)

References 10 publications

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“…For m 1 = m 2 = · · · = m n = 1, the above result was already known to Mongelli [12]. For m 1 = m 2 = · · · = m n = 1, the above result was already known to Mongelli [12].…”

Section: Comparing With Equation (43) We Obtainmentioning

confidence: 71%

On the descent polynomial of signed multipermutations

Zhao

2015

Proc. Amer. Math. Soc.

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Motivated by a conjecture of Savage and Visontai about the equidistribution of the descent statistic on signed permutations of the multiset {1, 1, 2, 2, . . . , n, n} and the ascent statistic on (1, 4, 3, 8, . . . , 2n − 1, 4n)inversion sequences, we investigate the descent polynomial of the signed permutations of a general multiset (multipermutations). We obtain a factorial generating function formula for a q-analog of these descent polynomials and apply it to show that they have only real roots. Two different proofs of the conjecture of Savage and Visontai are provided. Furthermore, multivariate identities that enumerate two different Euler-Mahonian distributions on type B Coxeter groups due to Beck and Braun are generalized to signed multipermutations.

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“…For m 1 = m 2 = · · · = m n = 1, the above result was already known to Mongelli [12]. For m 1 = m 2 = · · · = m n = 1, the above result was already known to Mongelli [12].…”

Section: Comparing With Equation (43) We Obtainmentioning

confidence: 71%

On the descent polynomial of signed multipermutations

Zhao

2015

Proc. Amer. Math. Soc.

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“…where exc A (w) is the number of excedances of w at coordinates of zero color and csum(w) is the sum of the colors of all coordinates of w; more explanation appears in Section 2. The flag excedance statistic was introduced (although not with this name) by Bagno and Garber [5] and was further studied in [17] and, in the special case r = 2, in [16,21]. We will denote by D r n the set of derangements (elements without fixed points of zero color) in Z r ≀ S n .…”

Section: Introduction and Resultsmentioning

confidence: 99%

Edgewise Subdivisions, Local $h$-Polynomials, and Excedances in the Wreath Product $\mathbb{Z}_r \wr \mathfrak{S}_n$

Athanasiadis¹

2014

SIAM J. Discrete Math.

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The coefficients of the local h-polynomial of the barycentric subdivision of the simplex with n vertices are known to count derangements in the symmetric group S n by the number of excedances. A generalization of this interpretation is given for the local h-polynomial of the rth edgewise subdivision of the barycentric subdivision of the simplex. This polynomial is shown to be γ-nonnegative and a combinatorial interpretation to the corresponding γ-coefficients is provided. The new combinatorial interpretations involve the notions of flag excedance and descent in the wreath product Z r ≀ S n . A related result on the derangement polynomial for Z r ≀ S n , studied by Chow and Mansour, is also derived from results of Linusson, Shareshian and Wachs on the homology of Rees products of posets.

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“…Bagno and Garber [2] introduced a definition of excedance on the set of signed permutations, called colored excedance, exc Clr(σ) := 2exc A (σ) + neg(σ). Mongelli [27] defined the generating function of the colored descent statistic on B n by P n (B, q) = σ∈Bn q exc Clr (σ) and proved P n (B, q) = (1 + q) n P n (A, q). (3.6) Let D n (B) denote the set of all derangements on B n and d n (B, q) be the generating function of the colored excedances on the set D n (B), that is…”

Section: The Eulerian Polynomials Of Type Bmentioning

confidence: 99%

Positivity and continued fractions from the binomial transformation

Zhu¹

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Given a sequence of polynomials $\{x_k(q)\}_{k \ges 0}$, define the transformation $$y_n(q) = a^n\sum\limits_{i = 0}^n {\left( \matrix{n \cr i} \right)} b^{n-i}x_i(q)$$ for $n\ges 0$. In this paper, we obtain the relation between the Jacobi continued fraction of the ordinary generating function of yn(q) and that of xn(q). We also prove that the transformation preserves q-TPr+1 (q-TP) property of the Hankel matrix $[x_{i+j}(q)]_{i,j \ges 0}$, in particular for r = 2,3, implying the r-q-log-convexity of the sequence $\{y_n(q)\}_{n\ges 0}$. As applications, we can give the continued fraction expressions of Eulerian polynomials of types A and B, derangement polynomials types A and B, general Eulerian polynomials, Dowling polynomials and Tanny-geometric polynomials. In addition, we also prove the strong q-log-convexity of derangement polynomials type B, Dowling polynomials and Tanny-geometric polynomials and 3-q-log-convexity of general Eulerian polynomials, Dowling polynomials and Tanny-geometric polynomials. We also present a new proof of the result of Pólya and Szegö about the binomial convolution preserving the Stieltjes moment property and a new proof of the result of Zhu and Sun on the binomial transformation preserving strong q-log-convexity.

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Excedances in classical and affine Weyl groups

Cited by 12 publications

References 10 publications

On the descent polynomial of signed multipermutations

On the descent polynomial of signed multipermutations

Edgewise Subdivisions, Local $h$-Polynomials, and Excedances in the Wreath Product $\mathbb{Z}_r \wr \mathfrak{S}_n$

Positivity and continued fractions from the binomial transformation

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