By considering the parity of the degrees and levels of nodes in increasing trees, a new combinatorial interpretation for the coefficients of the Taylor expansions of the Jacobi elliptic functions is found. As one application of this new interpretation, a conjecture of Ma-Mansour-Wang-Yeh is solved. Unifying the concepts of increasing trees and plane trees, Lin-Ma-Ma-Zhou introduced weakly increasing trees on a multiset. A symmetry joint distribution of "even-degree nodes on odd levels" and "odd-degree nodes" on weakly increasing trees is found, extending the Schett polynomials, a generalization of the Jacobi elliptic functions introduced by Schett, to multisets. A combinatorial proof and an algebraic proof of this symmetry are provided, as well as several relevant interesting consequences. Moreover, via introducing a group action on trees, we prove the partial γ-positivity of the multiset Schett polynomials, a result implies both the symmetry and the unimodality of these polynomials.

In this paper, we first give a sufficient condition for a sequence of polynomials to have alternatingly increasing property, and then we present a systematic study of excedance-type polynomials of permutations and derangements, signed or not, colored or not. Let p∈[0,1] and q∈[0,1] be two given real numbers. We prove that the cyc q-Eulerian polynomials of permutations are bi-γ-positive, and the fix and cyc (p,q)-Eulerian polynomials of permutations are alternatingly increasing, where fix and cyc are respectively the fixed point and cycle statistics. When q=1/2, we present a combinatorial interpretation of the bi-γ-coefficients of the cyc q-Eulerian polynomials. We then study excedance and flag excedance statistics of signed permutations and colored permutations. We establish relationships between the fix and cyc (p,q)-Eulerian polynomials and some multivariate excedance-type polynomials. In particular, we present a relationship between derangement polynomials of finite Coxeter groups of types A and D. Our results unify and generalize a variety of recent results. Moreover, one can see that the fix and cyc (p,q)-Eulerian polynomials of permutations contain a great deal of information about permutations and colored permutations.

Let ℐ_n,k be the set of k -colored involutions of order n and 𝒥_n,k be the set of k -colored involutions in ℐ_n,k without fixed points. Denote by des(π ,c) the number of descents of k -colored permutations (π ,c) . In this paper, it is proved that the following polynomials I_n,k(x)= ∑ _(π ,c)∈ℐ_n,kx^des(π ,c) ( n≥ 1, k≥ 1) and J_n,k(x)= ∑ _(π ,c)∈𝒥_n,kx^des(π ,c) ( n≥ 1, k≥ 2 ) are γ -positive.

Let M={1(p1),..., n(pn)} be a multiset, S-M,S-i be the set of multipermutations over M with i as the first entry and A(M,i)(x) be the enumerators of descents over S-M,S-i. A(M,i)(x) is called the (M, i)-multiset Eulerian polynomial. A Carlitz-type identity of A(M,i)(x) is derived. It is proved that xA(M,i)(x), c(1)A(M,i)(x) + c(2)A(M,j)(x) and c(1)xA(M,i)(x) + c(2)A(M,j)(x) have only real roots, where c(1) and c(2) are nonnegative real number, i, j is an element of M and i < j. For the multiset M={1(k), 2(k),..., n(k)}, it is shown that A(M,i)(x) is reciprocal with A(M, n-i+1)(x), A(M,i)(x) + A(M,n-i+1)(x) and xA(M,i)(x) + A(M,n-i+1)(x) are gamma-positive, and A(M,i)(x) is bi-gamma-positive for any 1 <= i <= n+1/2. For M={1, 2,..., n} and 1 <= i <= n, we give a combinatorial interpretation for gamma-coefficients of A(M,i)(x) + A(M, n-i+1)(x). (c) 2023 Elsevier Inc. All rights reserved.

In this paper, we introduce the definitions of Eulerian pair and Hermite-Biehler pair. We also characterize a duality relation between Eulerian recurrences and Eulerian recurrence systems. This generalizes and unifies Hermite-Biehler decompositions of several enumerative polynomials, including up-down run polynomials for symmetric groups, alternating run polynomials for hyperoctahedral groups, flag descent polynomials for hyperoctahedral groups and flag ascent-plateau polynomials for Stirling permutations. We derive some properties of associated polynomials. In particular, we prove the alternatingly increasing property and the interlacing property of the ascent-plateau and left ascent-plateau polynomials for Stirling permutations.

In this paper, we introduce the concept of weakly increasing trees on a multiset M, which is an extension of plane trees and increasing trees on the set {0, 1, ..., n}. We define the M-Eulerian-Narayana polynomial for weakly increasing trees on a multiset M, which interpolates between the Eulerian polynomial and the Narayana polynomial. We obtain a compact product formula for the number of weakly increasing trees on a general multiset. Inspired by some remarkable equidistributions between multipermutations and s-inversion sequences, we establish two connections between our M-Eulerian-Narayana polynomials for the multiset M = {1(2), 2(2), ..., n(2)}(resp. M = {1(2), 2(2), ..., (n - 1)(2), n}) and Savage and Schuster's s-Eulerian polynomials for the sequence s = (1, 1, 3, 2, 5, 3, ..., 2n - 1, n, n + 1)(resp. s = (1, 1, 3, 2, 5, 3, ..., 2n - 1, n)). We also derive equidistributions that involve some natural tree statistics among weakly increasing trees on different multisets. Via introducing a group action on weakly increasing trees, we prove combinatorially the gamma-positivity of the M-Eulerian-Narayana polynomials for a general multiset M. As an application of this gamma-positivity result, we obtain a combinatorial interpretation of gamma-coefficients of descent polynomials of permutations on the multiset {1(2), 2(2), ..., n(2)} in terms of weakly increasing trees. (C) 2021 Elsevier Inc. All rights reserved.

We prove that the enumerative polynomials of Stirling multipermutations by the statistics of plateaux, descents and ascents are partial γ-positive. Specialization of our result to the Jacobi-Stirling permutations confirms a recent partial γ-positivity conjecture due to Ma, Yeh and the second named author. Our partial γ-positivity expansion, as well as a combinatorial interpretation for the corresponding γ-coefficients, are obtained via the machine of context-free grammars and a group action on Stirling multipermutations. Besides, we also provide an alternative approach to the partial γ-positivity from the stability of certain multivariate polynomials on Stirling multipermutations. Moreover, we prove the partial γ-positivity for the enumerators of multipermutations by plateaux, descents and ascents via introducing a group action on words. Since multipermutations without any plateau are Smirnov words, our result generalizes a γ-positivity result due to Linusson, Shareshian and Wachs in this special case. Interestingly, our second action on multipermutations applies also to Stirling multipermutations and results in another combinatorial expansion for their partial γ-positivity. Finally, using a modification of our second group action and Foata's first fundamental transformation, we prove the partial γ-positivity for the enumerators of multipermutations by fixed points, excedances and drops, generalizing another result of Linusson, Shareshian and Wachs for derangements of a multiset.

Letters $x$ and $y$ alternate in a word $w$ if after deleting in $w$ all letters but the copies of $x$ and $y$ we either obtain a word $xyxy\cdots$ (of even or odd length) or a word $yxyx\cdots$ (of even or odd length). A graph $G=(V,E)$ is word-representable if and only if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy\in E$. It is known that a graph is word-representable if and only if it admits a certain orientation called semi-transitive orientation. Word-representable graphs generalize several important classes of graphs such as $3$-colorable graphs, circle graphs, and comparability graphs. There is a long line of research in the literature dedicated to word-representable graphs. However, almost nothing is known on word-representability of split graphs, that is, graphs in which the vertices can be partitioned into a clique and an independent set. In this paper, we shed a light to this direction. In particular, we characterize in terms of forbidden subgraphs word-representable split graphs in which vertices in the independent set are of degree at most 2, or the size of the clique is 4. Moreover, we give necessary and sufficient conditions for an orientation of a split graph to be semi-transitive.

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∈ [0,1] and q∈ [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.

The Jacobian elliptic function sn(u,k) is the inverse of the elliptic integral of the first kind and cn(u,k)=1−sn2(u,k). In this paper, we study coefficient polynomials in the Taylor series expansions of sn(u,k) and cn(u,k). We first provide a combinatorial expansion for a family of bivariate peak polynomials, which count permutations by their odd and even cycle peaks. A special case of this combinatorial expansion says that the coefficient polynomials of sn(u,k) are γ-positive. We then show that the coefficient polynomials of cn(u,k) are bi-γ-positive, which implies that these coefficient polynomials are unimodal with modes in the middle. Furthermore, by using context-free grammars, we find combinatorial interpretations of two associated coefficients in terms of increasing trees.

In this paper, we consider unimodal expansions of some multivariate Eulerian polynomials. We first give a sufficient condition for a polynomial to be alternatingly increasing. Let p ∈ [0, 1] and q ∈ [0, 1] be two given real numbers. We then prove that the cyc q-Eulerian polynomials of permutations are bi-γ-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. As applications, we discuss the bi-γ-positivity and alternatingly increasing property of several combinatorial polynomials, including 1/k-Eulerian polynomials, colored derangement polynomials and some refinements of the flag excedance polynomials. Finally, we study excedance statistics of colored permutations. In particular, we establish the relationships between some multivariate colored Eulerian polynomials and the (p, q)-Eulerian polynomials of permutations. Our results generalize several recent results in the literature.

We study two generalizations of the gamma-expansion of Eulerian polynomials from the viewpoint of the decompositions of statistics. We first present an expansion formula of the trivariate Eulerian polynomials, which are the enumerators for the joint distribution of descents, big ascents and successions of permutations. And then, inspired by the work of Chen and Fu on the trivariate second-order Eulerian polynomials, we show the e-positivity of the multivariate k-th order Eulerian polynomials, which are the enumerators for the joint distribution of ascents, descents and j-plateaux of k-Stirling permutations. We provide combinatorial interpretations for the coefficients of these two expansions in terms of increasing trees.

We prove that the enumerative polynomials of generalized Stirling permutations by the statistics of plateaux, descents and ascents are partial $γ$-positive. Specialization of our result to the Jacobi-Stirling permutations confirms a recent partial $γ$-positivity conjecture due to Ma, Yeh and the second named author. Our partial $γ$-positivity expansion, as well as a combinatorial interpretation for the corresponding $γ$-coefficients, are obtained via the machine of context-free grammars and a group action on generalized Stirling permutations. Besides, we also provide an alternative approach to the partial $γ$-positivity from the stability of certain multivariate polynomials.

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.

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.

In this paper, we first consider a generalization of the David-Barton identity which relate the alternating run polynomials to Eulerian polynomials. By using context-free grammars, we then present a combinatorial interpretation of a family of q-alternating run polynomials. Furthermore, we introduce the definition of semi-gamma-positive polynomial and we show the semi-gamma-positivity of the alternating run polynomials of dual Stirling permutations. A connection between the up-down run polynomials of permutations and the alternating run polynomials of dual Stirling permutations is established.

In this paper, we study γ-positivity of descent-type polynomials by introducing the change of context-free grammars method. We first present a unified grammatical proof of the γ-positivity of Eulerian polynomials, type B Eulerian polynomials, derangement polynomials, Narayana polynomials and type B Narayana polynomials. We then provide partial γ-positive expansions for several multivariate polynomials associated to Stirling permutations, Legendre-Stirling permutations, Jacobi-Stirling permutations and type B derangements. The recurrence relations for the partial γ-coefficients of these expansions are also obtained. By using some variants of the Foata-Strehl group action, we provide combinatorial interpretations for the coefficients of most of these partial γ-positive expansions.

The circular descent of a permutation a is a set {sigma(i) vertical bar sigma(i) > sigma(i + 1)}. In this paper, we focus on the enumerations of permutations by the circular descent set. Let cdes(n) (S) be the number of permutations of length n which have the circular descent set S. We derive the explicit formula for cdes(n )(S). We describe a class of generating binary trees T-k with weights. We find that the number of permutations in the set CDESn (S) corresponds to the weights of T-k. As a application of the main results in this paper, we also give the enumeration of permutation tableaux according to their shape.