Adopting the definition of excedances of type B due to Brenti, we give a type B analogue of the q-derangement polynomials. The connection between q-derangement polynomials and Eulerian polynomials naturally extends to the type B case. Based on this relation, we derive some basic properties of the q-derangement polynomials of type B, including the generating function formula, the Sturm sequence property, and the asymptotic normal distribution. We also show that the q-derangement polynomials are almost symmetric in the sense that the coefficients possess the spiral property.
We consider several generalizations of the classical $γ$-positivity of Eulerian polynomials (and their derangement analogues) using generating functions and combinatorial theory of continued fractions. For the symmetric group, we prove an expansion formula for inversions and excedances as well as a similar expansion for derangements. We also prove the $γ$-positivity for Eulerian polynomials for derangements of type $B$. More general expansion formulae are also given for Eulerian polynomials for $r$-colored derangements. Our results answer and generalize several recent open problems in the lite…
The derangement polynomial for the symmetric group enumerates derangements by the number of excedances. It can be interpreted as the local $h$-polynomial, in the sense of Stanley, of the barycentric subdivision of the simplex. Motivated by this interpretation, we define a derangement polynomial for the even-signed permutation group. The coefficients of this polynomial are nonnegative, symmetric and unimodal. We show that they enumerate derangements in the even-signed permutation group according to a notion of excedance, which is analogous to the one introduced by Brenti for signed permutations…
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 $\mathfrak{S}_n$ by the number of excedances. A generalization of this interpretation is given for the local $h$-polynomial of the $r$th 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 wre…
The classical Eulerian polynomials $A_n(t)$ are known to be gamma positive. Define the positive Eulerian polynomial $\mathsf{AExc^{+}}_n(t)$ as the polynomial obtained when we sum excedances over the alternating group. We show that $\mathsf{AExc^{+}}_n(t)$ is gamma positive iff $n \geq 5$ and $n \equiv 1$ (mod 2). When $n \geq 4$, and $n \equiv 0$ (mod 2) we show that $\mathsf{AExc^{+}}_n(t)$ can be written as a sum of two gamma positive polynomials. Similar results are shown when we consider the positive type-D and type-D Eulerian polynomials. Finally, we show gamma positivity results when we…
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 $p\in [0,1]$ and $q\in [0,1]$ 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 incre…
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 th…
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 Stembridge's 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 Zhuang's generalizations [Adv. in Appl. Math. 90 (2017) 86-144] using descent statistics of permutations. Our methods include permu…
The derangement polynomial $d_n (x)$ for the symmetric group enumerates derangements by the number of excedances. The derangement polynomial $d^B_n(x)$ for the hyperoctahedral group is a natural type $B$ analogue. A new combinatorial formula for this polynomial is given in this paper. This formula implies that $d^B_n (x)$ decomposes as a sum of two nonnegative, symmetric and unimodal polynomials whose centers of symmetry differ by a half and thus provides a new transparent proof of its unimodality. A geometric interpretation, analogous to Stanley's interpretation of $d_n (x)$ as the local $h$-…
We provide combinatorial interpretation for the $γ$-coefficients of the basic Eulerian polynomials that enumerate permutations by the excedance statistic and the major index as well as the corresponding $γ$-coefficients for derangements. Our results refine the classical $γ$-positivity results for the Eulerian polynomials and the derangement polynomials. The main tools are Brändén's modified Foata--Strehl action on permutations and the recent triple statistic (des, rix,aid) equidistibuted with (exc, fix, maj).
We consider properties of the box polynomials, a one variable polynomial defined over all integer partitions $λ$ whose Young diagrams fit in an $m$ by $n$ box. We show that these polynomials can be expressed by the finite difference operator applied to the power $x^{m+n}$. Evaluating box polynomials yields a variety of identities involving set partition enumeration. We extend the latter identities using restricted growth words and a new operator called the fast Fourier operator, and consider connections between set partition enumeration and the chromatic polynomial on graphs. We also give conn…
In this research announcement we present a new q-analog of a classical formula for the exponential generating function of the Eulerian polynomials. The Eulerian polynomials enumerate permutations according to their number of descents or their number of excedances. Our q-Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index. There is a vast literature on q-Eulerian polynomials which involve other combinations of Mahonian and Eulerian permutation statistics, but the combination of major index and excedance number seems to have been com…
It is a classical result that the parity-balance of the number of weak excedances of all permutations (derangements, respectively) of length $n$ is the Euler number $E_n$, alternating in sign, if $n$ is odd (even, respectively). Josuat-Vergès obtained a $q$-analog of the results respecting the number of crossings of a permutation. One of the goals in this paper is to extend the results to the permutations (derangements, respectively) of types B and D, on the basis of the joint distribution in statistics excedances, crossings and the number of negative entries obtained by Corteel, Josuat-Vergès…
The linearization coefficient $\mathcal{L}(L_{n_1}(x)\dots L_{n_k}(x))$ of classical Laguerre polynomials $L_n(x)$ is known to be equal to the number of $(n_1,\dots,n_k)$-derangements, which are permutations with a certain condition. Kasraoui, Stanton and Zeng found a $q$-analog of this result using $q$-Laguerre polynomials with two parameters $q$ and $y$. Their formula expresses the linearization coefficient of $q$-Laguerre polynomials as the generating function for $(n_1,\dots,n_k)$-derangements with two statistics counting weak excedances and crossings. In this paper their result is proved …
We consider a sequence of four variable polynomials by refining Stieltjes' continued fraction for Eulerian polynomials. Using combinatorial theory of Jacobi-type continued fractions and bijections we derive various combinatorial interpretations in terms of permutation statistics for these polynomials, which include special kinds of descents and excedances in a recent paper of Baril and Kirgizov. As a by-product, we derive several equidistribution results for permutation statistics, which enables us to confirm and strengthen a recent conjecture of Vajnovszki and also to obtain several compagnio…
This paper was motivated by a conjecture of Brändén (European J. Combin. \textbf{29} (2008), no.~2, 514--531) about the divisibility of the coefficients in an expansion of generalized Eulerian polynomials, which implies the symmetric and unimodal property of the Eulerian numbers. We show that such a formula with the conjectured property can be derived from the combinatorial theory of continued fractions. We also discuss an analogous expansion for the corresponding formula for derangements and prove a $(p,q)$-analogue of the fact that the (-1)-evaluation of the enumerator polynomials of permuta…
In this note, by the umbra calculus method, the Sun and Zagier's congruences involving the Bell numbers and the derangement numbers are generalized to the polynomial cases. Some special congruences are also provided.
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 $k\geq 1/2$. In particular, we get a symmetric decomposition of $d_n^B(x;1/2)$ in terms of the classical derangement polynomials…
In this paper we introduce the definition of marked permutations. We first present a bijection between Stirling permutations and marked permutations. We then present an involution on Stirling derangements. Furthermore, we present a symmetric bivariate enumerative polynomials on $r$-colored marked permutations. Finally, we give an explanation of $r$-colored marked permutations by using the language of combinatorial objects.
We use a probabilistic approach to describe the behavior as $n -> \infty$ of the Laplace transforms of $P^n$, where $P$ a fixed complex polynomial. As a consequence we obtain a new elementary proof of an result of Gillis-Ismail-Offer in the combinatorial theory of derangements.
The Eulerian polynomials and derangement polynomials are two well-studied generating functions that frequently arise in combinatorics, algebra, and geometry. When one makes an appearance, the other often does so as well, and their corresponding generalizations are similarly linked. This is this case in the theory of subdivisions of simplicial complexes, where the Eulerian polynomial is an $h$-polynomial and the derangement polynomial is its local $h$-polynomial. Separately, in Ehrhart theory the Eulerian polynomials are generalized by the $h^\ast$-polynomials of $s$-lecture hall simplices. Her…
A classical result states that the parity balance of the number of excedances of all permutations (derangements, respectively) of length $n$ is the Euler number. In 2010, Josuat-Vergès gives a $q$-analogue with $q$ representing the number of crossings. We extend this result to the permutations (derangements, respectively) of type B and D. It turns out that the signed countings are related to the derivative polynomials of $\tan$ and $\sec$. Springer numbers defined by Springer can be regarded as an analogue of Euler numbers defined on every Coxeter group. In 1992 Arnol'd showed that the Springe…
In this paper, we study gamma-positivity of descent-type polynomials by introducing the change of context-free grammars method. We first present grammatical proofs of the gamma-positivity of the Eulerian polynomials, type B Eulerian polynomials, derangement polynomials, Narayana polynomials and type B Narayana polynomials. We then provide partial gamma-positive expansions for several multivariate polynomials associated to Stirling permutations, Legendre-Stirling permutations, Jacobi-Stirling permutations and type B derangements, and the recurrences for the partial gamma-coefficients of these e…
The descent polynomials of separable permutations and derangements are both demonstrated to be unimodal. Moreover, we prove that the $γ$-coefficients of the first are positive with an interpretation parallel to the classical Eulerian polynomial, while the second is spiral, a property stronger than unimodality. Furthermore, we conjecture that they are both real-rooted.
In this note, using the derangement polynomials and their umbral representation, we give another simple proof of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory.
In algebraic, topological, and geometric combinatorics inequalities among the coefficients of combinatorial polynomials are frequently studied. Recently a notion called the alternatingly increasing property, which is stronger than unimodality, was introduced. In this paper, we relate the alternatingly increasing property to real-rootedness of the symmetric decomposition of a polynomial to develop a systematic approach for proving the alternatingly increasing property for several classes of polynomials. We apply our results to strengthen and generalize real-rootedness, unimodality, and alternat…
The (p,q)-Analogues of Bi-Stirling Eulerian Polynomials via continued fractions
Universite Lyon 1（里昂第一大学）；UMR 5208 du CNRS, Institut Camille Jordan（法国国家科学研究中心第5208联合研究实验室，卡米尔·让研究所）；College of Mathematics and Physics, Wenzhou University（温州大学数学与物理学院）；Université Claude Bernard Lyon 1（克洛德·贝尔纳里昂大学）；CNRS UMR 5208, Institut Camille Jordan（法国国家科学研究中心第5208联合研究实验室，卡米尔·让研究所）
Eulerian-type polynomials over matchings and matching permutations
School of Mathematics and Statistics, Shandong University of Technology（山东理工大学数学与统计学院）；Department of Mathematics and Statistics, University of Strathclyde（斯特拉斯克莱德大学数学与统计系）；Department of Mathematics, National Kaohsiung Normal University（国立高雄师范大学数学系）；College of Mathematics and Physics, Wenzhou University（温州大学数理学院）
