γ-positivity and partial γ-positivity of descent-type polynomials

Ma, Shi-Mei; Ma, Jun; Yeh, Yeong‐Nan

doi:10.1016/j.jcta.2019.05.002

Cited by 32 publications

(41 citation statements)

References 41 publications

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“…with the initial conditions F 0 (x, y; k) = 1 and F 1 (x, y; k) = x + y. By rewriting (20) in terms of the generating function F := F (x, y, z; k), we have…”

Section: Letmentioning

confidence: 99%

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The $1/k$-Eulerian Polynomials of Type $B$

Yeh

et al. 2020

Electron. J. Combin.

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In this paper, we define the $1/k$-Eulerian polynomials of type $B$. Properties of these polynomials, including combinatorial interpretations, recurrence relations and $\gamma$-positivity are studied. In particular, we show that the $1/k$-Eulerian polynomials of type $B$ are $\gamma$-positive when $k>0$. Moreover, we define the $1/k$-derangement polynomials of type $B$, denoted $d_n^B(x;k)$. We show that the polynomials $d_n^B(x;k)$ are bi-$\gamma$-positive when $k\geq 1/2$. In particular, we get a symmetric decomposition of the polynomials $d_n^B(x;1/2)$ in terms of the classical derangement polynomials.

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“…with the initial conditions F 0 (x, y; k) = 1 and F 1 (x, y; k) = x + y. By rewriting (20) in terms of the generating function F := F (x, y, z; k), we have…”

Section: Letmentioning

confidence: 99%

“…Combining (20) and (22), it is routine to verify (7). (ii) We now consider a change of the grammar G given in Lemma 13.…”

Section: Letmentioning

confidence: 99%

The $1/k$-Eulerian Polynomials of Type $B$

Yeh

et al. 2020

Electron. J. Combin.

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“…which can be regarded as an extension to the second order Eulerian polynomials of Gessel The question of γ-positivity for C n (x, y, z) has been studied by Ma-Ma-Yeh [17]. Write…”

Section: Introductionmentioning

confidence: 99%

“…The partial γ-positivity for several multivariate ploynomials associated various classes of permutations has recently been studied in Ma-Ma-Yeh [17], Lin-Ma-Zhang [13] and Yan-Huang-Yang [18].…”

Section: Introductionmentioning

confidence: 99%

“…The objective of this paper is to present a combinatorial treatment of the partial γcoefficients of C M (x, y, z). In particular, our strategy can be adapted to deal with the partial γ-coefficients of the second order Eulerian polynomials, which in turn can be readily converted to the combinatorial formulation obtained by Ma-Ma-Yeh [17]. Our combinatorial interpretation of γ M,i,j is built on the Gessel trees which are increasing plane trees in which the internal vertices are represented by distinct numbers, whereas in the combinatorial framework of Yan, Huang and Yang [18], two vertices are allowed to be represented by the same number.…”

Section: Introductionmentioning

confidence: 99%

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The Gessel Correspondence and the Partial $γ$-Positivity of the Eulerian Polynomials on Multiset Stirling Permutations

Chen¹,

Fu²,

Yan³

2021

Preprint

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Pondering upon the grammatical labeling of 0-1-2 increasing plane trees, we came to the realization that the grammatical labels play a role as records of chopped off leaves of the original increasing binary trees. While such an understanding is purely psychological, it does give rise to an efficient apparatus to tackle the partial γ-positivity of the Eulearian polynomials on multiset Stirling permutations, as long as we bear in mind the combinatorial meanings of the labels x and y in the Gessel representation of a k-Stirling permutation by means of an increasing (k + 1)-ary tree. More precisely, we introduce a Foata-Strehl action on the Gessel trees resulting in an interpretation of the partial γ-coefficients of the aforementioned Eulerian polynomials, different from the ones found by Lin-Ma-Zhang and Yan-Huang-Yang. In particular, our strategy can be adapted to deal with the partial γcoefficients of the second order Eulerian polynomials, which in turn can be readily converted to the combinatorial formulation due to Ma-Ma-Yeh in connection with certain statistics of Stirling permutations.

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The Dumont Ansatz for the Eulerian Polynomials, Peak Polynomials and Derivative Polynomials

Chen

Fu²

2022

Ann. Comb.

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γ-positivity and partial γ-positivity of descent-type polynomials

Cited by 32 publications

References 41 publications

The $1/k$-Eulerian Polynomials of Type $B$

The $1/k$-Eulerian Polynomials of Type $B$

The Gessel Correspondence and the Partial $γ$-Positivity of the Eulerian Polynomials on Multiset Stirling Permutations

The Dumont Ansatz for the Eulerian Polynomials, Peak Polynomials and Derivative Polynomials

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