In this paper, we study the generating functions for the number of pattern restricted Stirling permutations with a given number of plateaus, descents and ascents. Properties of the generating functions, including symmetric properties and explicit formulas are studied. Combinatorial explanations are given for some equidistributions.
We define a variation of Stirling permutations, called quasi-Stirling permutations, to be permutations on the multiset $\{1,1,2,2,\ldots, n,n\}$ that avoid the patterns 1212 and 2121. Their study is motivated by a known relationship between Stirling permutations and increasing ordered rooted labeled trees. We construct a bijection between quasi-Stirling permutations and the set of ordered rooted labeled trees and investigate pattern avoidance for these permutations.
Stirling permutations were introduced by Gessel and Stanley, who used their enumeration by the number of descents to give a combinatorial interpretation of certain polynomials related to Stirling numbers. Quasi-Stirling permutations, which can be viewed as labeled noncrossing matchings, were introduced by Archer et al. as a natural extension of Stirling permutations. Janson's correspondence between Stirling permutations and labeled increasing plane trees extends to a bijection between quasi-Stirling permutations and the same set of trees without the increasing restriction. Archer et al. posed …
In this note we generalize an identity of John Riordan and Robert Donaghey relating the enumerator for Stirling permutations to the Eulerian polynomials.
We prove that the enumerative polynomials of quasi-Stirling permutations of multisets with respect to the statistics of plateaux, descents and ascents are partial $γ$-positive, thereby confirming a recent conjecture posed by Lin, Ma and Zhang. This is accomplished by proving the partial $γ$-positivity of the enumerative polynomials of certain ordered labeled trees, which are in bijection with quasi-Stirling permutations of multisets. As an application, we provide an alternative proof of the partial $γ$-positivity of the enumerative polynomials on Stirling permutations of multisets.
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…
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. In particular, by using Foata and Strehl's group action, we prove two bistatistics over the set of Stirling permutations of order n are equidistributed.
The development of the theories of the second-order Eulerian polynomials began with the works of Buckholtz and Carlitz in their studies of an asymptotic expansion. Gessel-Stanley introduced Stirling permutations and presented combinatorial interpretations of the second-order Eulerian polynomials. Recently, there is a growing interest in the properties of Stirling permutations. The motivation of this paper is to develop a general method for finding equidistributed statistics on Stirling permutations. Firstly, we show that the up-down-pair statistic is equidistributed with ascent-plateau statist…
A permutation can be locally classified according to the four local types: peaks, valleys, double rises and double falls. The corresponding classification of binary increasing trees uses four different types of nodes. Flajolet demonstrated the continued fraction representation of the generating function of local types, using a classical bijection between permutations, binary increasing trees, and suitably defined path diagrams induced by Motzkin paths. The aim of this article is to extend the notion of local types from permutations to $k$-Stirling permutations (also known as $k$-multipermutati…
We present (bi-)symmetric generating functions for the joint distributions of Euler-Stirling statistics on permutations, including the number of descents ($\mathsf{des}$), inverse descents ($\mathsf{ides}$), the number of left-to-right maxima ($\mathsf{lmax}$), the number of right-to-left maxima ($\mathsf{rmax}$) and the number of left-to-right minima ($\mathsf{lmin}$). We also show how they recover the classical symmetric generating function of permutations due to Carlitz, Roselle and Scoville (1966). Our proofs exploit three different recursive constructions of inversion sequences, bijection…
We present exponential generating function analogues to two classical identities involving the ordinary generating function of the complete homogeneous symmetric functions. After a suitable specialization the new identities reduce to identities involving the first and second order Eulerian polynomials. The study of these identities led us to consider a family of symmetric functions associated with a class of permutations introduced by Gessel and Stanley, known in the literature as Stirling permutations. In particular, we define certain type statistics on Stirling permutations that refine the s…
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…
Pondering upon the grammatical labeling of 0-1-2 increasing plane trees, we come to the realization that the grammatical labels play a role as records of chopped off leaves of the original increasing binary trees. While such an understanding is purely psychological, it does give rise to an efficient apparatus to tackle the partial $γ$-positivity of the Eulearian polynomials on multiset Stirling permutations, as long as we bear in mind the combinatorial meanings of the labels $x$ and $y$ in the Gessel representation of a $k$-Stirling permutation by means of an increasing $(k+1)$-ary tree. More …
A permutation $π$ of a multiset is said to be a {\em quasi-Stirling } permutation if there does not exist four indices $i<j<k<\ell$ such that $π_i=π_k$ and $π_j=π_{\ell}$. Define $$ \overline{Q}_{\mathcal{M}}(t,u,v)=\sum_{π\in \overline{\mathcal{Q}}_{\mathcal{M}}}t^{des(π)}u^{asc(π)}v^{plat(π)},$$ where $\overline{\mathcal{Q}}_{\mathcal{M}}$ denotes the set of quasi-Stirling permutations on the multiset $\mathcal{M}$, and $asc(π)$ (resp. $des(π)$, $plat(π)$) denotes the number of ascents (resp. descents, plateaux) of $π$. Denote by $\mathcal{M}^σ$ the multiset $\{1^{σ_1}, 2^{σ_2}, \ldots, n^{σ…
In this note we investigate mixed partitions with extra condition on the sizes of the blocks. We give a general formula and the generating function. We consider in more details a special case, determining the generating functions, some recurrences and a connection to r-Stirling numbers. To obtain our results, we use pure combinatorial arguments, classical manipulations of generating functions and to derive the generating functions we apply the symbolic method.
In this paper we introduce mixed coloured permutation, permutations with certain coloured cycles, and study the enumerative properties of these combinatorial objects. We derive the generating function, closed forms, recursions and combinatorial identities for the counting sequence, mixed Stirling numbers of the first kind. In this comprehensive study we consider further the conditions on the length of the cycles, $r$-mixed Stirling numbers and the connection to Bell polynomials.
Generalized Stirling permutations, families of increasing trees and urn models
Bona [2007+]研究了Gessel和Stanley [1978]引入的Stirling排列类中上升、平稳和下降的分布。最近，Janson [2008+]证明了Stirling置换和平面递归树之间的联系，并证明了Bona考虑的参数的联合正规律。这里我们将考虑广义Stirling置换，推广了Bona和Janson的早期结果，并将它们与某些广义平面递归树族以及$(k+1)$ -元增树联系起来。我们还给出了两个不同的双射之间的某些家庭的增长树，这两个作为一个特殊情况的双射之间的三元增长树和平面递归树。为了描述感兴趣的参数的（渐近）行为，我们研究了三个（广义）Polya瓮模型使用各种方法。
We study set partitions with $r$ distinguished elements and block sizes found in an arbitrary index set $S$. The enumeration of these $(S,r)$-partitions leads to the introduction of $(S,r)$-Stirling numbers, an extremely wide-ranging generalization of the classical Stirling numbers and the $r$-Stirling numbers. We also introduce the associated $(S,r)$-Bell and $(S,r)$-factorial numbers. We study fundamental aspects of these numbers, including recurrence relations and determinantal expressions. For $S$ with some extra structure, we show that the inverse of the $(S,r)$-Stirling matrix encodes th…
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.
Restricted Whitney numbers of the first kind appear in the combinatorial recursion for the matroid Kazhdan-Lusztig polynomials. In the special case of braid matroids (the matroid associated to the partition lattice, the complete graph, the type A Coxeter arrangement and the symmetric group) these restricted Whitney numbers are Stirling numbers of the first kind. We use this observation to obtain a formula for the coefficients of the Kazhdan-Lusztig polynomials for braid matroids in terms of sums of products of Stirling numbers of the first kind. This results in new identities between Stirling …
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…
Given a general multiset $\mathcal{M}=\{1^{m_1},2^{m_2},\ldots,n^{m_n}\}$, where $i$ appears $m_i$ times, a multipermutation $π$ of $\mathcal{M}$ is called {\em quasi-Stirling}, if it contains no subword of the form $abab$ with $a\neq b$. We designate exactly one entry of $π$, say $k\in \mathcal{M}$, which is not the leftmost entry among all entries with the same value, by underlining it in $π$, and we refer to the pair $(π,k)$ as a quasi-Stirling multipermutation of $\mathcal{M}$ rooted at $k$. By introducing certain vertex and edge labeled trees, we give a new bijective proof of an identity …
Using Reiner's definition of Stirling numbers of the second kind in types $B$ and $D$, we generalize two well-known identities concerning the classical Stirling numbers of the second kind. The first identity relates them with Eulerian numbers and the second identity interprets them as entries in a transition matrix between the elements of two standard bases of the polynomial ring $R[x]$. Finally, we generalize these identities to the group of colored permutations $G_{m,n}$.
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）
