The notion of a descent polynomial, a function in enumerative combinatorics that counts permutations with specific properties, enjoys a revived recent research interest due to its connection with other important notions in combinatorics, viz. peak polynomials and symmetric functions. We define the function $\mathfrak{d}^{m}(I,n)$ as a generalization of the descent polynomial and we prove that for any positive integer $m$, this function is a polynomial in $n$ for sufficiently large $n$ (similarly to the descent polynomial). We obtain an explicit formula for $\mathfrak{d}^{m}(I,n)$ when $m$ is sufficiently large. We look at the coefficients of $\mathfrak{d}^{m}(I,n)$ in different falling factorial bases. We prove the positivity of the coefficients and discover a combinatorial interpretation for them. This result is similar to the positivity result of Diaz-Lopez et al. for the descent polynomial.
We prove a combinatorial formula for the alternating descent polynomials of type A and B. Combining with Josuat-Vergès' combinatorial interpretation for Hoffman's derivative polynomials for tangent and secant functions, we obtain a unified combinatorial proof of Ma-Yeh's two formulae linking these two families of polynomials.
We investigate the $α$-colored Eulerian polynomials and a notion of descents introduced in a recent paper of Hedmark and show that such polynomials can be computed as a polynomial encoding descents computed over a quotient of the wreath product $\mathbb{Z}_α\wr\mathfrak{S}_n$. Moreover, we consider the flag descent statistic computed over this same quotient and find that the flag Eulerian polynomial remains palindromic. We prove that the flag descent polynomial is palindromic over this same quotient by giving a combinatorial proof that the flag descent statistic is symmetrically distributed ov…
We prove several identities expressing polynomials counting permutations by various descent statistics in terms of Eulerian polynomials, extending results of Stembridge, Petersen, and Brändén. Additionally, we find $q$-exponential generating functions for $q$-analogues of these descent statistic polynomials that also keep track of the inversion number or inverse major index. We also present identities relating several of these descent statistic polynomials to refinements of type B Eulerian polynomials and flag descent polynomials by the number of negative letters of a signed permutation. Our m…
We introduce the notion of the descent set polynomial as an alternative way of encoding the sizes of descent classes of permutations. Descent set polynomials exhibit interesting factorization patterns. We explore the question of when particular cyclotomic factors divide these polynomials. As an instance we deduce that the proportion of odd entries in the descent set statistics in the symmetric group S_n only depends on the number on 1's in the binary expansion of n. We observe similar properties for the signed descent set statistics.
Descent polynomials and peak polynomials, which enumerate permutations with given descent and peak sets respectively, have recently received considerable attention. We give several formulas for $q$-analogs of these polynomials which refine the enumeration by the length of the permutations. In the case of $q$-descent polynomials we prove that the coefficients in one basis are strongly $q$-log concave, and conjecture this property in another basis. For peaks, we prove that the $q$-peak polynomial is palindromic in $q$, resolving a conjecture of Diaz-Lopez, Harris, and Insko.
The descent polynomial of a finite $I\subseteq \mathbb{Z}^+$ is the polynomial $d(I,n)$, for which the evaluation at $n>\max(I)$ is the number of permutations on $n$ elements, such that $I$ is the set of indices where the permutation is descending. In this paper we will prove some conjectures concerning coefficient sequences of $d(I,n)$. As a corollary we will describe some zero-free regions for the descent polynomial.
In this paper we present an explicit formula for the number of permutations with a given number of alternating descents. Moreover, we study the interlacing property of the real parts of the zeros of the generating polynomials of these numbers.
Motivated by a conjecture of Savage and Visontai about the equidistribution of the descent statistic on signed permutations of the multiset $\{1,1,2,2,\ldots,n,n\}$ and the ascent statistic on $(1,4,3,8,\ldots,2n-1,4n)$-inversion sequences, we investigate the descent polynomial of the signed permutations of a general multiset. We obtain a factorial generating function formula for a $q$-analog of these descent polynomials and apply it to show that they have only real roots. Two different proofs of the conjecture of Savage and Visontai are provided.
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.
Bóna conjectured that the descent polynomials on $(n-2)$-stack sortable permutations have only real zeros. Brändén proved this conjecture by establishing a more general result. In this paper, we give another proof of Brändén's result by using the theory of $s$-Eulerian polynomials recently developed by Savage and Visontai.
For a poset whose Hasse diagram is a rooted plane forest $F$, we consider the corresponding tree descent polynomial $A_F(q)$, which is a generating function of the number of descents of the labelings of $F$. When the forest is a path, $A_F(q)$ specializes to the classical Eulerian polynomial. We prove that the coefficient sequence of $A_F(q)$ is unimodal and that if $\{T_{n}\}$ is a sequence of trees with $|T_{n}| = n$ and maximal down degree $D_{n} = O(n^{0.5-ε})$ then the number of descents in a labeling of $T_{n}$ is asymptotically normal.
We prove a conjecture by Diaz-Lopez et al. that bounds the roots of descent polynomials. To do so, we prove an algebraic inequality, which we refer to as the "Slice and Push Inequality." This inequality compares expressions that come from Naruse's hook-length formula for the number of standard Young tableaux of a skew shape.
Let $n$ be a nonnegative integer and $I$ be a finite set of positive integers. In 1915, MacMahon proved that the number of permutations in the symmetric group $\mathfrak{S}_n$ with descent set $I$ is a polynomial in $n$. We call this the descent polynomial. However, basic properties of these polynomials such as a description of their coefficients and roots do not seem to have been studied in the literature. Much more recently, in 2013, Billey, Burdzy, and Sagan showed that the number of elements of $\mathfrak{S}_n$ with peak set $I$ is a polynomial in $n$ times a certain power of two. Since th…
Motivated by juggling sequences and bubble sort, we examine permutations on the set {1,2,...,n} with d descents and maximum drop size k. We give explicit formulas for enumerating such permutations for given integers k and d. We also derive the related generating functions and prove unimodality and symmetry of the coefficients.
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 classical Eulerian polynomials $A_n(t)$ are known to be gamma positive. Define the positive Eulerian polynomial $A_n^+(t)$ as the polynomial obtained when we sum descents over the alternating group. We show that $A_n^+(t)$ is gamma positive iff $n \equiv 0,1$ (mod 4). When $n \equiv 2$ (mod 4) we show that $A_n^+(t)$ can be written as a sum of two gamma positive polynomials while if $n \equiv 3$ (mod 4), we show that $A_n^+(t)$ can be written as a sum of three gamma positive polynomials. Similar results are shown when we consider the positive type-D and type-D Eulerian polynomials.
Motivated by the work of Chung, Claesson, Dukes, and Graham, we define a natural type B analog of the classic bubble sort, and use it to define a type B analog of the maximum drop statistic. We enumerate (by explicit, recursive, and generating function formulas) signed permutations with r type B descents and type B maximum drop at most k. We also find a connection between these signed permutations and certain 2-colored juggling sequences.
In this paper we study the generating polynomials obtained by enumerating signed simsun permutations by number of the descents. Properties of the polynomials, including the recurrence relations and generating functions are studied.
In this paper we examine the sorting operator $T(LnR)=T(R)T(L)n$. Applying this operator to a permutation is equivalent to passing the permutation reversed through a stack. We prove theorems that characterise $t$-revstack sortability in terms of patterns in a permutation that we call $zigzag$ patterns. Using these theorems we characterise those permutations of length $n$ which are sorted by $t$ applications of $T$ for $t=0,1,2,n-3,n-2,n-1$. We derive expressions for the descent polynomials of these six classes of permutations and use this information to prove Steingrímsson's sorting conjecture…
Motivated by recent results on quasi-Stirling permutations, which are permutations of the multiset $\{1,1,2,2,\dots,n,n\}$ that avoid the "crossing" patterns 1212 and 2121, we consider nonnesting permutations, defined as those that avoid the patterns 1221 and 2112 instead. We show that the polynomial giving the distribution of the number of descents on nonnesting permutations is a product of an Eulerian polynomial and a Narayana polynomial. It follows that, rather unexpectedly, this polynomial is palindromic. We provide bijective proofs of these facts by composing various transformations on Dy…
We continue to explore cyclotomic factors in the descent set polynomial $Q_{n}(t)$, which was introduced by Chebikin, Ehrenborg, Pylyavskyy and Readdy. We obtain large classes of factors of the form $Φ_{2s}$ or $Φ_{4s}$ where $s$ is an odd integer, with many of these being of the form $Φ_{2p}$ where $p$ is a prime. We also show that if $Φ_{2}$ is a factor of $Q_{2n}(t)$ then it is a double factor. Finally, we give conditions for an odd prime power $q = p^{r}$ for which $Φ_{2p}$ is a double factor of $Q_{2q}(t)$ and of $Q_{q+1}(t)$.
We study Naruse-Newton coefficients, which are obtained from expanding descent polynomials in a Newton basis introduced by Jiradilok and McConville. These coefficients $C_0, C_1, \ldots$ form an integer sequence associated to each finite set of positive integers. For fixed nonnegative integers $a<b$, we examine the set $R_{a, b}$ of all ratios $\frac{C_a}{C_b}$ over finite sets of positive integers. We characterize finite sets for which $\frac{C_a}{C_b}$ is minimized and provide a construction to prove $R_{a, b}$ is unbounded above. We use this construction to obtain results on the closure of …
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…
We study a group action on permutations due to Foata and Strehl and use it to prove that the descent generating polynomial of certain sets of permutations has a nonnegative expansion in the basis $\{t^i(1+t)^{n-1-2i}\}_{i=0}^m$, $m=\lfloor (n-1)/2 \rfloor$. This property implies symmetry and unimodality. We prove that the action is invariant under stack-sorting which strengthens recent unimodality results of Bóna. We prove that the generalized permutation patterns $(13-2)$ and $(2-31)$ are invariant under the action and use this to prove unimodality properties for a $q$-analog of the Eulerian …
The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. We show that the alternating descent polynomials on permutations are unimodal via a five-term recurrence relation. We also found a quadratic recursion for the alternating major index $q$-analog of the alternating descent polynomials. As an interesting application of this quadratic recursion, we show that $(1+q)^{\lfloor n/2\rfloor}$ divides $\sum_{π\in\mathfrak{S}_n}q^{\rm{altmaj}(π)}$, where $\mathfrak{S}_n$ is the set of all permutations of $\{1,2,\ldots,n\}$ and $\rm{altmaj}(π…
A Smirnov word is a word over the positive integers in which adjacent letters must be different. A symmetric function enumerating these words by descent number arose in the work of Shareshian and the second named author on $q$-Eulerian polynomials, where a $t$-analog of a formula of Carlitz, Scoville, and Vaughan for enumerating Smirnov words is proved. A symmetric function enumerating a circular version of these words by cyclic descent number arose in the work of the first named author on chromatic quasisymmetric functions of directed graphs, where a $t$-analog of a formula of Stanley for enu…
In the study of Kostka numbers and Catalan numbers, Kirillov posed a unimodality conjecture for the rectangular Narayana polynomials. We prove that the rectangular Narayana polynomials have only real zeros, and thereby confirm Kirillov's unimodality conjecture with the help of Newton's inequality. By using an equidistribution property between descent numbers and ascent numbers on ballot paths due to Sulanke and a bijection between lattice words and standard Young tableaux, we show that the rectangular Narayana polynomial is equal to the descent generating function on standard Young tableaux of…
We introduce new natural generalizations of the classical descent and inversion statistics for permutations, called width-$k$ descents and width-$k$ inversions. These variations induce generalizations of the excedance and major statistics, providing a framework in which the most well-known equidistributivity results for classical statistics are paralleled. We explore additional relationships among the statistics providing specific formulas in certain special cases. Moreover, we explore the behavior of these width-$k$ statistics in the context of pattern avoidance.
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 …
We define some generalizations of the classical descent and inversion statistics on signed permutations that arise from the work of Sack and Ulfarsson [20] and called after width-k descents and width-k inversionsof type A in Davis's work [8]. Using the aforementioned new statistics, we derive some new generalizations of Eulerian polynomials of type A, B and D. It should also be noticed that we establish the Gamma-positivity of the "width-k" Eulerian polynomials and we give a combinatorial interpretation of finite sequences associated to these new polynomials using quasisymmetric functions and …
Lusztig's fake degree is the generating polynomial for the major index of standard Young tableaux of a given shape. Results of Springer and James & Kerber imply that, mysteriously, its evaluation at a $k$-th primitive root of unity yields the number of border strip tableaux with all strips of size $k$, up to sign. This is essentially the special case of the Murnaghan-Nakayama rule for evaluating an irreducible character of the symmetric group at a rectangular partition. We refine this result to standard Young tableaux and border strip tableaux with a given number of descents. To do so, we intr…
We consider the inversion enumerator I_n(q), which counts labeled trees or, equivalently, parking functions. This polynomial has a natural extension to generalized parking functions. Substituting q = -1 into this generalized polynomial produces the number of permutations with a certain descent set. In the classical case, this result implies the formula I_n(-1) = E_n, the number of alternating permutations. We give a combinatorial proof of these formulas based on the involution principle. We also give a geometric interpretation of these identities in terms of volumes of generalized chain polyto…
A result of Foata and Schutzenberger states that two statistics on permutations, the number of inversions and the inverse major index, have the same distribution on a descent class. We give a multivariate generalization of this property: the sorted vectors of the Lehmer code, of the inverse majcode, and of a new code (the inverse saillance code), have the same distribution on a descent class, and their common multivariate generating function is a flagged ribbon Schur function.
In this paper, we consider the number of occurrences of descents, ascents, 123-subwords, 321-subwords, peaks and valleys in flattened permutations, which were recently introduced by Callan in his study of finite set partitions. For descents and ascents, we make use of the kernel method and obtain an explicit formula (in terms of Eulerian polynomials) for the distribution on $\mathcal{S}_n$ in the flattened sense. For the other four patterns in question, we develop a unified approach to obtain explicit formulas for the comparable distributions. We find that the formulas so obtained for 123- and…
Recently, Lazar and Wachs (arXiv:1910.07651) showed that the (median) Genocchi numbers play a fundamental role in the study of the homogenized Linial arrangement and obtained two new permutation models (called D-permutations and E-permutations) for (median) Genocchi numbers. They further conjecture that the distributions of cycle numbers over the two models are equal. In a follow-up, Eu et al. (arXiv:2103.09130) further proved the gamma-positivity of the descent polynomials of even-odd descent permutations, which are in bijection with E-permutations by Foata's fundamental transformation. This …
Given an rc-graph $R$ of permutation $w$ and an rc-graph $Y$ of permutation $v$, we provide an insertion algorithm, which defines an rc-graph $R\leftarrow Y$ in the case when $v$ is a shuffle with the descent at $r$ and $w$ has no descents greater than $r$ or in the case when $v$ is a shuffle, whose shape is a hook. This algorithm gives a combinatorial rule for computing the generalized Littlewood-Richardson coefficients $c^{u}_{wv}$ in the two cases mentioned above.
Motivated by the work of Visontai and Dey-Sivasubramanian on the gamma-positivity of some polynomials, we find the commutative property of a pair of Eulerian operators. As an application, we show the bi-gamma-positivity of the descent polynomials on permutations of the multiset $\{1^{a_1},2^{a_2},\ldots,n^{a_n}\}$, where $0\leqslant a_i\leqslant 2$. Therefore, these descent polynomials are all alternatingly increasing, and so they are unimodal with modes in the middle.
The notion of descent set, for permutations as well as for standard Young tableaux (SYT), is classical. Cellini introduced a natural notion of {\em cyclic descent set} for permutations, and Rhoades introduced such a notion for SYT --- but only for rectangular shapes. In this work we define {\em cyclic extensions} of descent sets in a general context, and prove existence and essential uniqueness for SYT of almost all shapes. The proof applies nonnegativity properties of Postnikov's toric Schur polynomials, providing a new interpretation of certain Gromov-Witten invariants.
In 1916, MacMahon showed that permutations in $S_n$ with a fixed descent set $I$ are enumerated by a polynomial $d_I(n)$. Diaz-Lopez, Harris, Insko, Omar, and Sagan recently revived interest in this descent polynomial, and suggested the direction of studying such enumerative questions for other consecutive patterns (descents being the consecutive pattern $21$). Zhu studied this question for the consecutive pattern $321$. We continue this line of work by studying the case of any consecutive pattern of the form $k,k-1,\ldots,1$, which we call a $k$-descent. In this paper, we reduce the problem o…
Let $W\ltimes L$ be an irreducible affine Weyl group with Coxeter complex $Σ$, where $W$ denotes the associated finite Weyl group and $L$ the translation subgroup. The Steinberg torus is the Boolean cell complex obtained by taking the quotient of $Σ$ by the lattice $L$. We show that the ordinary and flag $h$-polynomials of the Steinberg torus (with the empty face deleted) are generating functions over $W$ for a descent-like statistic first studied by Cellini. We also show that the ordinary $h$-polynomial has a nonnegative $γ$-vector, and hence, symmetric and unimodal coefficients. In the class…
Dalian University of Technology（大连理工大学）；Institute of Mathematical Sciences, Dalian University of Technology（大连理工大学数学科学学院）；LSEC, Inst. Comp. Math., Academy of Mathematics and System Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学研究所/科学计算与数值分析实验室）
The aim of this paper is to make a systematical study on the stability of polynomials in combinatorics. Applying the characterizations of Borcea and Brändén concerning linear operators preserving stability, we present criteria for real stability and Hurwitz stability. We also give a criterion for Hurwitz stability of the Turán expressions. As applications, we derive some stability results occurred in the literature in a unified manner. In addition, we obtain the Hurwitz stability of Turán expressions for alternating runs polynomials of types $A$ and $B$ and solve a conjecture concerning Hurwit…
