Descent Polynomials Research Articles

Connecting descent and peak polynomials

A permutation $\sigma=\sigma_1 \sigma_2 \cdots \sigma_n$ has a descent at $i$ if $\sigma_i>\sigma_{i+1}$. A descent $i$ is called a peak if $i>1$ and $i-1$ is not a descent. The size of the set of all permutations of $n$ with a given descent set is a polynomials in $n$, called the descent polynomial. Similarly, the size of the set of all permutations of $n$ with a given peak set, adjusted by a power of $2$ gives a polynomial in $n$, called the peak polynomial. In this work we give a unitary expansion of descent polynomials in terms of peak polynomials. Then we use this expansion to give an interpretation of the coefficients of the peak polynomial in a binomial basis, thus giving a constructive proof of the peak polynomial positivity conjecture.

Real stable polynomials and the alternatingly increasing property

In this paper, we first present a criterion for the real stability of recursive polynomials by applying the characterization of Borcea and Brändén concerning linear operators preserving stability. As applications, we derive some real stability results occurred in the literature in a unified manner. Moreover, we prove the real-rootedness of peak polynomials in bicolored dyck paths and ternary polynomials.Second, based on the h-polynomials from combinatorial geometry, we use real stability to get two criteria for interlacing property of zeros between a polynomial and its reciprocal polynomial, which in particular imply the alternatingly increasing property of the original polynomial or its reciprocal polynomial. These criteria extend a result of Brändén and Solus and unify such properties for many combinatorial polynomials, including ascent polynomials for k-ary words, descent polynomials on signed Stirling permutations and colored permutations, q-analog of descent polynomials on colored permutations and descent polynomials on color-signed permutations. Furthermore, we also obtain a recurrence relation and zero-interlacing property of q-analog of descent polynomials on colored permutations that extend some results of Brändén and Brenti. In addition, as an application of weak Hurwitz stability, we get the alternatingly increasing property and zero-interlacing property for two kinds of peak polynomials on the dual set of Stirling permutations.

Roots of Descent Polynomials and an Algebraic Inequality on Hook Lengths


 
 
 By reinterpreting the descent polynomial as a function enumerating standard Young tableaux of a ribbon shape, we use Naruse's hook-length formula to express the descent polynomial as a product of two polynomials: one is a trivial part which is a product of linear factors, and the other comes from the excitation factor of Naruse's formula. We expand the excitation factor positively in a Newton basis which arises naturally from Naruse's formula. Under this expansion, each coefficient is the weight of a certain combinatorial object, which we introduce in this paper. We introduce and prove the "Slice and Push Inequality", which compares the weights of such combinatorial objects. As a consequence, we establish a proof of a conjecture by Diaz-Lopez et al. that bounds the roots of descent polynomials.
 
 

Some γ-positive polynomials arising from enumerations of the pseudo Schröder paths

We introduce a notion of pseudo Schröder paths, which are counted by the Schröder numbers. Using the symbolic method and the Lagrange inversion formula, we provide an alternative proof that the descent polynomials of separable permutations are γ-positive. We also present several classes of γ-positive polynomials via enumerating the pseudo Schröder paths according to several combinatorial statistics by applying Riordan arrays.

Commuting Eulerian operators

Motivated by the work of Visontai and Dey–Sivasubramanian on the gamma-positivity of some polynomials, we discover 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 {1a1,2a2,…,nan}, where 0⩽ai⩽2. Therefore, these descent polynomials are all alternatingly increasing, and so they are unimodal with modes in the middle.

Descent polynomials for labeled trees

Descent polynomials for labeled trees

Enumeration of permutations by the parity of descent positions

Noticing that some recent variations of descent polynomials are special cases of Carlitz and Scoville's four-variable polynomials, which enumerate permutations by the parity of descent and ascent positions, we prove a q-analogue of Carlitz-Scoville's generating function by counting the inversion number and a type B analogue by enumerating the signed permutations with respect to the parity of descent and ascent positions. As a by-product of our formulas, we obtain a q-analogue of Chebikin's formula for alternating descent polynomials, an alternative proof of Sun's gamma-positivity of her bivariate Eulerian polynomials and a type B analogue, the latter refines Petersen's gamma-positivity of the type B Eulerian polynomials.

Cycles of even-odd drop permutations and continued fractions of Genocchi numbers

Recently Lazar and Wachs proved two new permutation models, called D-permutations and E-permutations, for Genocchi and median Genocchi numbers. In a follow-up, Eu et al. studied the even-odd descent permutations, which are in bijection with E-permutations. We generalize Eu et al.'s descent polynomials with eight statistics and obtain an explicit J-fraction formula for their ordinary generaing function. The J-fraction permits us to confirm two conjectures of Lazar-Wachs about cycles of D and E permutations and obtain a (p,q)-analogue of Eu et al.'s gamma-formula. Moreover, the (p,q) gamma-coefficients have the same factorization flavor as the gamma-coefficients of Brändén's (p,q)-Eulerian polynomials.

A generalization of descent polynomials

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 dm(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 dm(I,n) when m is sufficiently large. We look at the coefficients of dm(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.

On the [formula omitted]-positivity of multiset Eulerian polynomials

Let An(p)(t) (resp. Bn(p)(t)) be the descent polynomial on permutations of the multiset {1p,2p,…,np} (resp. {1p,2p,…,np,n+1}). The γ-positivity of An(p)(t) was known but to give a combinatorial interpretation for the corresponding γ-coefficients still remains open. In this paper we• manage to find a combinatorial interpretation for the γ-coefficients of An(p)(t) via the model of weakly increasing trees;• and prove that Bn(p)(t) has bi-γ-positivity expansion Bn(p)(t)=an(t)+tbn(t), where bn(t)=(p−1)An(p)(t).The first result, whose proof makes use of Chen’s general bijective algorithm for trees and a new decomposition of weakly increasing trees, answers a recent open problem posed by Lin–Ma–Ma–Zhou. The latter result, which we provide both a computational proof and a short combinatorial proof, extends a bi-γ-positivity result due to Ma–Ma–Yeh from p=2 to general p. Their proof for p=2 uses Chen’s context-free grammar and seems hard to be adopted for general p.

Eulerian pairs and Eulerian recurrence systems

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.

Some coefficient sequences related to the descent polynomial

The descent polynomial of a finite I⊆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.

On [formula omitted]-analogs of descent and peak polynomials

Descent polynomials and peak polynomials, which enumerate permutations π∈Sn with given descent and peak sets respectively, have recently received considerable attention (Billey et al., 1989;Diaz-Lopez et al., 2019). We give several formulas for q-analogs of these polynomials which refine the enumeration by the length of π. In the case of q-descent polynomials we prove that the coefficients in one basis are stronglyq-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 et al. (2021).

Weakly increasing trees on a multiset

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={12,22,…,n2} (resp. M={12,22,…,(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 γ-positivity of the M-Eulerian–Narayana polynomials for a general multiset M. As an application of this γ-positivity result, we obtain a combinatorial interpretation of γ-coefficients of descent polynomials of permutations on the multiset {12,22,…,n2} in terms of weakly increasing trees.

On the last fall degree of Weil descent polynomial systems

Given a polynomial system F over a finite field k which is not necessarily of dimension zero, we consider the Weil descent F′ of F over a subfield k′. We prove a theorem which relates the last fall degrees of F1 and F1′, where the zero set of F1 corresponds bijectively to the set of k-rational points of F, and the zero set of F1′ is the set of k′-rational points of the Weil descent F′. As an application we derive upper bounds on the last fall degree of F1′ in the case where F is a set of linearized polynomials.

Eulerian polynomials and excedance statistics

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 (Zhuang 2017 [29]) using descent statistics of permutations. Our methods include permutation enumeration techniques involving variations of classical bijections from permutations to Laguerre histories, explicit continued fraction expansions of combinatorial generating functions in Shin and Zeng (2012) [21] and cycle version of modified Foata-Strehl action. We also prove similar formulae for restricted permutations such as derangements and permutations avoiding certain patterns. Moreover, we provide new combinatorial interpretations for the γ-coefficients of the inversion polynomials restricted on 321-avoiding permutations.

Tree Descent Polynomials: Unimodality and Central Limit Theorem

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 $$D_{n} = O(n^{0.5-\epsilon }),$$ then the number of descents in a labeling of $$T_{n}$$ is asymptotically normal.

Descent polynomials

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 Sn 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 Sn with peak set I is a polynomial in n times a certain power of two. Since then, there have been a flurry of papers investigating properties of this peak polynomial. The purpose of the present paper is to study the descent polynomial. We will see that it displays some interesting parallels with its peak relative. Conjectures and questions for future research are scattered throughout.

On two unimodal descent polynomials

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.

A sextuple equidistribution arising in Pattern Avoidance

We construct an intriguing bijection between 021-avoiding inversion sequences and (2413,4213)-avoiding permutations, which proves a sextuple equidistribution involving double Eulerian statistics. Two interesting applications of this result are also presented. Moreover, this result inspires us to characterize all permutation classes that avoid two patterns of length 4 whose descent polynomial equals that of separable permutations.