Jun Ma

• Shanghai Jiao Tong University

Let \({\mathcal {I}}_{n,k}\) be the set of k-colored involutions of order n and \({\mathcal {J}}_{n,k}\) be the set of k-colored involutions in \({\mathcal {I}}_{n,k}\) without fixed points. Denote by \(des(\pi ,c)\) the number of descents of k-colored permutations \((\pi ,c)\). In this paper, it is proved that the following polynomials $$\begin{al...

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, alternati...

The Jacobian elliptic function $\sn(u,k)$ is the inverse of the elliptic integral of the first kind and $\cn(u,k)=\sqrt{1-\sn^2(u,k)}$. 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 permu...

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...

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 comp...

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 grammatica...

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 no...

In this paper, we characterize a duality relation between Eulerian recurrences and Eulerian recurrence systems, which generalizes and unifies Hermite-Biehler decompositions of several enumerative polynomials, including flag descent polynomials for hyperoctahedral group, flag ascent-plateau polynomials for Stirling permutations, up-down run polynomi...

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 po...

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 earlie...

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 resu...

Let s = (s 1 , s 2 ,. .. , s n ,. . .) be a sequence of positive integers. An s-inversion sequence of length n is a sequence e = (e 1 , e 2 ,. .. , e n) of nonnegative integers such that 0 e i < s i for 1 i n. When s i = (i − 1)k + 1 for any i 1, we call the s-inversion sequences the k-inversion sequences. In this paper, we provide a bijective proo...

A polynomial p(z) of degree d is alternatingly increasing if and only if it can be decomposed into a sum p(z)=a(z)+zb(z), where a(z) and b(z) are symmetric and unimodal polynomials with dega(z)=d and degb(z)<d. We say that p(z) is bi-gamma-positive if a(z) and b(z) are both gamma positive. In this paper, we present a unified elementary proof of the...

In this paper, we study gamma-positivity of descent-type polynomials by introducing the change of context-free grammars method. We first present a unified grammatical proof of the gamma-positivity of Eulerian polynomials, type B Eulerian polynomials, derangement polynomials, Narayana polynomials and type B Narayana polynomials. We then provide part...

In this paper, we first consider an alternate formulation of the David-Barton identity which relates the alternating run polynomials to Eulerian polynomials. By using this alternate formulation, we see that for any γ-positive polynomial, there exists a David-Barton type identity. We then consider the joint distribution of cycle runs and cycles over...

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 a...

In this paper, several variants of the ascent-plateau statistic are introduced, including flag ascent-plateau, double ascent and descent-plateau. We first study the flag ascent-plateau statistic on Stirling permutations by using context-free grammars. We then present a unified refinement of the ascent polynomials and the ascent-plateau polynomials....

The Legendre-Stirling numbers of the second kind were introduced by Everitt et al. in the spectral theory of powers of the Legendre differential expressions. As a continuation of the work of Andrews and Littlejohn (Proc. Amer. Math. Soc., 137 (2009), 2581-2590), we provide a combinatorial code for Legendre-Stirling set partitions. As an application...

The Jacobian elliptic function $\sn(u,k)$ is the inverse of the elliptic integral of the first kind and $\cn(u,k)=\sqrt{1-\sn^2(u,k)}$. 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 permu...

The Legendre-Stirling numbers of the second kind were introduced by Everitt et al. in the spectral theory of powers of the Legendre differential expressions. In this paper, we provide a combinatorial code for Legendre-Stirling set partitions. As an application, we obtain combinatorial expansions of the Legendre-Stirling numbers of both kinds. Moreo...

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 g...

In this paper, several variants of the ascent-plateau statistic are introduced, including flag ascent-plateau, double ascent and descent-plateau. We first study the flag ascent-plateau statistic on Stirling permutations by using context-free grammars. We then present a unified refinement of the ascent polynomials and the ascent-plateau polynomials....

Binomial-Eulerian polynomials were introduced by Postnikov, Reiner and Williams. In this paper, properties of the binomial-Eulerian polynomials, including recurrence relations and generating functions are studied. We present three constructive proofs of the recurrence relations for binomial-Eulerian polynomials.

In this paper, we present grammatical descriptions of several polynomials associated with Eulerian polynomials, including q-Eulerian polynomials, alternating run polynomials and derangement polynomials. As applications, we get several convolution formulas involving these polynomials.

In this paper, we present grammatical descriptions of several polynomials associated with Eulerian polynomials, including q-Eulerian polynomials, alternating run polynomials and derangement polynomials. As applications, we get several convolution formulas involving these polynomials.

In this paper, let Δ be a nonsingular M-matrix. A generalization of G-parking functions, which is called Δ-parking functions, is studied. An explicit characterization for Δ-parking functions is given. It is shown that Δ-parking functions can be obtained by a simple way from recurrent configurations on the nonsingular M-matrix Δ. It is proved that t...

In this paper we study the cycle descent statistic on permutations. Several involutions on permutations and derangements are constructed. Moreover, we construct a bijection between negative cycle descent permutations and Callan perfect matchings.

In this paper we study the cycle descent statistic on permutations. Several involutions on permutations and derangements are constructed. Moreover, we construct a bijection between negative cycle descent permutations and Callan perfect matchings.

In this paper, a bivariate generating function \(CF(x,y) = \frac{{f(x) - yf(xy)}} {{1 - y}}\) is investigated, where f(x) = Σn⩾0f n x n is a generating function satisfying the functional equation f(x) = 1 + Σ j=1r Σ i=j−1m a ij x i f(x)j . In particular, we study lattice paths in which their end points are on the line y = 1. Rooted lattice paths ar...

A partition of a given set is said to be uniform if all the partition classes have the same cardinality. In this paper, we will introduce the concepts of rooted -lattice path and rooted cyclic permutation and prove some fundamental theorems concerning the actions of rooted cyclic permutations on rooted lattice -paths. The main results obtained have...

Let $\Delta$ be an integer $n \times n$-matrix which satisfies the conditions: $\det \Delta\neq 0$, $\Delta_{ij}\leq 0\text{ for }i\neq j,$ and there exists a vector ${\bf r}=(r_1,\ldots,r_n)>0$ such that ${\bf r}\Delta \geq 0$. Here the notation ${\bf r}> 0$ means that $r_i>0$ for all $i$, and ${\bf r}\geq {\bf r}'$ means that $r_i\geq r'_i$ for e...

In this paper, we determine the necessary and sufficient conditions for the existence of simple incomplete triple systems for all λ ≤ 6. incomplete triple system, simple, embedding, difference triple.

The classical Chung-Feller theorem (2) tells us that the number of Dyck paths of length n with m ∞aws is the n-th Catalan number and independent on m. In this paper, we consider the reflnements of Dyck paths with ∞aws by four parameters, namely peak, valley, double descent and double ascent. Let pn;m;k be the number of all the Dyck paths of semi-le...

Let G be a connected and simple graph with vertex set {1, 2, …, n + 1} and TG(x, y) the Tutte polynomial of G. In this paper, we give combinatorial interpretations for TG(1, −1). In particular, we give the definitions of even spanning tree and left spanning tree. We prove TG(1, −1) is the number of even-left spanning trees of G. We associate a perm...

In this paper, we study the combinatorial properties of w-IPP (identifiable parents property) codes and give necessary and sufficient conditions for a code to be a w-IPP code. Furthermore, let R(C) = 1/n logq |C| denote the rate of the q-ary code C of length n, suppose q ≥ 3 is a prime power, we prove that there exists a sequence of linear q-ary 2-...

The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with $m$ flaws is the $n$-th Catalan number and independent on $m$. L. Shapiro [9] found the Chung-Feller properties for the Motzkin paths. Mohanty's book [5] devotes an entire section to exploring Chung-Feller theorem. Many Chung-Feller theorems are consequ...

The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with flaws $m$ is the $n$-th Catalan number and independent on $m$. L. Shapiro [7] found the Chung-Feller properties for the Motzkin paths. In this paper, we find the connections between these two Chung-Feller theorems. We focus on the weighted versions of t...

Let $G$ be a connected graph with vertex set $\{0,1,2,...,n\}$. We allow $G$ to have multiple edges and loops. In this paper, we give a characterization of external activity by some parameters of $G$-parking functions. In particular, we give the definition of the bridge vertex of a $G$-parking function and obtain an expression of the Tutte polynomi...

The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with $m$ flaws is the $n$-th Catalan number and independent on $m$. In this paper, we consider the refinements of Dyck paths with flaws by four parameters, namely peak, valley, double descent and double ascent. Let ${p}_{n,m,k}$ be the number of all the Dyck...

The conceptions of $G$-parking functions and $G$-multiparking functions were introduced in [15] and [12] respectively. In this paper, let $G$ be a connected graph with vertex set $\{1,2,...,n\}$ and $m\in V(G)$. We give the definition of $(G,m)$-multiparking function. This definition unifies the conceptions of $G$-parking function and $G$-multipark...

In this paper, we focus on Dyck paths with peaks and valleys, avoiding an arbitrary set of heights. The generating functions of such types of Dyck paths can be represented by continued fractions. We also discuss a special case that requires all peak and valley heights to avoid congruence classes modulo k. We study the shift equivalence on sequences...

The circular descent of a permutation $\sigma$ is a set $\{\sigma(i)\mid \sigma(i)>\sigma(i+1)\}$. 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...

The circular peak set of a permutation $\sigma$ is the set $\{\sigma(i)\mid \sigma(i-1)<\sigma(i)>\sigma(i+1)\}$. Let $\mathcal{P}_n$ be the set of all the subset $S\subseteq [n]$ such that there exists a permutation $\sigma$ which has the circular set $S$. We can make the set $\mathcal{P}_n$ into a poset $\mathscr{P}_n$ by defining $S\preceq T$ if...

The circular peak set of a permutation $\sigma$ is the set $\{\sigma(i)\mid \sigma(i-1)<\sigma(i)>\sigma(i+1)\}$. In this paper, we focus on the enumeration problems for permutations by circular peak sets. Let $cp_n(S)$ denote the number of the permutations of order $n$ which have the circular peak set $S$. For the case with $|S|=0,1,2$, we derive...

In this paper, we focus on ordered $k$-flaw preference sets. Let $\mathcal{OP}_{n,\geq k}$ denote the set of ordered preference sets of length $n$ with at least $k$ flaws and $\mathcal{S}_{n,k}=\{(x_1,...,x_{n-k})\mid x_1+x_2+... +x_{n-k}=n+k, x_i\in\mathbb{N}\}$. We obtain a bijection from the sets $\mathcal{OP}_{n,\geq k}$ to $\mathcal{S}_{n,k}$....

In this paper, let $\mathcal{P}_{n;\leq s;k}^l$ denote a set of $k$-flaw preference sets $(a_1,...,a_n)$ with $n$ parking spaces satisfying that $1\leq a_i\leq s$ for any $i$ and $a_1=l$ and $p_{n;\leq s;k}^l=|\mathcal{P}_{n;\leq s;k}^l|$. We use a combinatorial approach to the enumeration of $k$-flaw preference sets by their leading terms. The app...

In this paper, let $\mathcal{P}_{n,n+k;\leq n+k}$ (resp. $\mathcal{P}_{n;\leq s}$) denote the set of parking functions $\alpha=(a_1,...,a_n)$ of length $n$ with $n+k$ (respe. $n$)parking spaces satisfying $1\leq a_i\leq n+k$ (resp. $1\leq a_i\leq s$) for all $i$. Let $p_{n,n+k;\leq n+k}=|\mathcal{P}_{n,n+k;\leq n+k}|$ and $p_{n;\leq s}=|\mathcal{P}...