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Register a new userThe 1/k-Eulerian polynomials of type B
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In this paper, we give a type B analogue of the 1/k-Eulerian polynomials. Properties of this kind of 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 obtain the corresponding results for derangements of type B. We show that a type B 1/k-derangement polynomials $d_n^B(x;k)$ are bi-gamma-positive when $kgeq 1/2$. In particular, we get a symmetric decomposition of $d_n^B(x;1/2)$ in terms of the classical derangement polynomials.
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Inspired by the recent work of Chen and Fu on the e-positivity of trivariate second-order Eulerian polynomials, we show the e-positivity of a family of multivariate k-th order Eulerian polynomials. A relationship between the coefficients of this e-positive expansion and second-order Eulerian numbers is established. Moreover, we present a grammatical proof of the fact that the joint distribution of the ascent, descent and j-plateau statistics over k-Stirling permutations are symmetric distribution. By using symmetric transformation of grammars, a symmetric expansion of trivariate Schett polynomial is also established.
In this paper we present grammatical interpretations of the alternating Eulerian polynomials of types A and B. As applications, we derive several properties of the type B alternating Eulerian polynomials, including combinatorial expansions, recurrence relations and generating functions. We establish an interesting connection between alternating Eulerian polynomials of type B and left peak polynomials of permutations in the symmetric group, which implies that the type B alternating Eulerian polynomials have gamma-vectors alternate in sign.
A formula of Stembridge states that the permutation peak polynomials and descent polynomials are connected via a quadratique transformation. The aim of this paper is to establish the cycle analogue of Stembridges formula by using cycle peaks and excedances of permutations. We prove a series of new general formulae expressing polynomials counting permutations by various excedance statistics in terms of refined Eulerian polynomials. Our formulae are comparable with Zhuangs generalizations [Adv. in Appl. Math. 90 (2017) 86-144] using descent statistics of permutations. Our methods include permutation enumeration techniques involving variations of classical bijections from permutations to Laguerre histories, explicit continued fraction expansions of combinatorial generating functions in Shin and Zeng [European J. Combin. 33 (2012), no. 2, 111--127] and cycle version of modified Foata-Strehl action. We also prove similar formulae for restricted permutations such as derangements and permutations avoiding certain patterns. Moreover, we provide new combinatorial interpretations for the $gamma$-coefficients of the inversion polynomials restricted on $321$-avoiding permutations.
We prove that the Eulerian polynomial satisfies certain polynomial congruences. Furthermore, these congruences characterize the Eulerian polynomial.
The binomial Eulerian polynomials, introduced by Postnikov, Reiner, and Williams, are $gamma$-positive polynomials and can be interpreted as $h$-polynomials of certain flag simplicial polytopes. Recently, Athanasiadis studied analogs of these polynomials for colored permutations. In this paper, we generalize them to $mathbf{s}$-inversion sequences and prove that these new polynomials have only real roots by the method of interlacing polynomials. Three applications of this result are presented. The first one is to prove the real-rootedness of binomial Eulerian polynomials, which confirms a conjecture of Ma, Ma, and Yeh. The second one is to prove that the symmetric decomposition of binomial Eulerian polynomials for colored permutations is real-rooted. Thirdly, our polynomials for certain $mathbf{s}$-inversion sequences are shown to admit a similar geometric interpretation related to edgewise subdivisions of simplexes.
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