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Register a new userThe gamma-positivity of Eulerian polynomials and succession statistics
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This paper is concerned with multivariate refinements of the gamma-positivity of Eulerian polynomials by using the succession and fixed point statistics. Properties of the enumerative polynomials for permutations, signed permutations and derangements, including generating functions and gamma-positivity are studied, which generalize and unify earlier results of Athanasiadis, Brenti, Chow, Petersen, Roselle, Stembridge, Shin and Zeng. In particular, we derive a formula expressing the joint distribution of excedance number and negative number statistics over the type B derangements in terms of the derangement polynomials.
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We provide combinatorial interpretation for the $gamma$-coefficients of the basic Eulerian polynomials that enumerate permutations by the excedance statistic and the major index as well as the corresponding $gamma$-coefficients for derangements. Our results refine the classical $gamma$-positivity results for the Eulerian polynomials and the derangement polynomials. The main tools are Brandens modified Foata--Strehl action on permutations and the recent triple statistic (des, rix,aid) equidistibuted with (exc, fix, maj).
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.
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.
The object of this paper is to give a systematic treatment of excedance-type polynomials. We first give a sufficient condition for a sequence of polynomials to have alternatingly increasing property, and then we present a systematic study of the joint distribution of excedances, fixed points and cycles of permutations and derangements, signed or not, colored or not. Let $pin [0,1]$ and $qin [0,1]$ be two given real numbers. We prove that the cyc q-Eulerian polynomials of permutations are bi-gamma-positive, and the fix and cyc (p,q)-Eulerian polynomials of permutations are alternatingly increasing, and so they are unimodal with modes in the middle, where fix and cyc are the fixed point and cycle statistics. When p=1 and q=1/2, we find a combinatorial interpretation of the bi-gamma-coefficients of the (p,q)-Eulerian polynomials. We then study excedance and flag excedance statistics of signed permutations and colored permutations. In particular, we establish the relationships between the (p,q)-Eulerian polynomials and some multivariate Eulerian polynomials. Our results unify and generalize a variety of recent results.
Ma-Ma-Yeh made a beautiful observation that a change of the grammar of Dumont instantly leads to the $gamma$-positivity of the Eulearian polynomials. We notice that the transformed grammar bears a striking resemblance to the grammar for 0-1-2 increasing trees also due to Dumont. The appearance of the factor of two fits perfectly in a grammatical labeling of 0-1-2 increasing plane trees. Furthermore, the grammatical calculus is instrumental to the computation of the generating functions. This approach can be adapted to study the $e$-positivity of the trivariate second-order Eulerian polynomials introduced by Janson, in connection with the joint distribution of the numbers of ascents, descents and plateaux over Stirling permutations.
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