Some coefficient sequences related to the descent polynomial

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Abstract

The descent polynomial of a finite IZ+ 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.

Introduction

Denote the group of permutations on [n]={1,,n} by Sn. For a permutation πSn, the set of descending positions is Des(π)={i[n1]πi>πi+1}.We would like to investigate the number of permutations in Sn with a fixed descent set. More precisely, for a finite IZ+, let m=max(I{0}). Then for n>m, we can count the number of permutations with descent set I that we will denote by d(I,n)=|D(I,n)|=|{πSnDes(π)=I}|.

In 1915, this function was shown to be a degree m polynomial in n by MacMahon [6]. The descents of permutations gained popularity after Solomon [8] proved that the sums of permutations with the same descent set form a linear basis for a subalgebra of the group algebra called the Solomon algebra (the result is true for any Coxeter group when one extends the definition of descents). Then, in 2013, Billey, Burdzy, and Sagan [1] proved combinatorial results about a similar notion called peaks of permutations. For a permutation πSn the set of indexes Peak(π)={i[2,n1]πi1<πi>πi+1} are the peak positions. They have proved that the counting function P(S,n)=|{πSnPeak(π)=S}|is a product of a polynomial pS(n) and the exponential 2n#S1. Also in the same paper, they proposed some conjectures about the coefficients of pS(n) in some bases. The zeros and coefficients of the peak polynomial were further investigated by Billy, Fahrbach and Talmage in [2] and by Diaz-Lopez et al. in [4]. In 2019, Diaz-Lopez et al. [5] used similar techniques to find some recursions of descent polynomials. In this paper, we answer several of their conjectures.

The coefficient sequence ak(I) is defined uniquely through the following equation: d(I,n)=k=0mak(I)nmk.In [5], it was shown that the sequence ak(I) is nonnegative since it counts some combinatorial objects. By taking a transformation of this sequence, we were able to apply Stanley’s theorem about the statistics of heights of a fixed element in a poset. As a result, we prove

Theorem 3.4

If I, then the sequence {ak(I)}k=0m is log-concave, that means that for any 0<k<m we have ak1(I)ak+1(I)ak2(I).

As a byproduct of the proof of Theorem 3.4 we get a better understanding of the coefficients, therefore we are able to obtain the following bound on the roots of d(I,n):

Theorem 5.3

If I and d(I,z0)=0 for some z0, then |z0|m.

As in [5] we will also consider the ck(I) coefficient sequence, that is defined by the following equation d(I,n)=k=0m(1)mkck(I)n+1k.By using a new recursion from [5] we prove that

Proposition 4.3

If I, then for any 0km the coefficient ck(I)0.

In the last section, we will establish zero-free regions for descent polynomials. In particular, we will prove the following.

Theorem 5.10

If I and d(I,z0)=0 for some z0, then |z0m|m+1. In particular, z01, where z0 is the real part of z0.

This paper is organized as follows. In the next section we will define two sequences, ak(I) and ck(I), recall the two main recursions for the descent polynomial, and introduce one of our main key ingredients from Stanley [9], [10] about the log-concavity of the height polynomial of a fix element in a poset. In Section 3, we will prove a conjecture concerning the sequence ak(I). In Section 4, we will prove a conjecture concerning the sequence ck(I) and some consequences. In Section 5, we prove some bounds on the roots of the descent polynomials that will rely on a result of ak(I) and some further investigation of the sequence ck(I).

Section snippets

Preliminaries

In this section, we will recall some recursions of the descent polynomial associated with a set I and we will establish some related coefficient sequences by choosing different bases for the polynomials.

First of all, for the rest of the paper we will always denote a finite subset of Z+ by I, the maximal element of I{0} by m(I). If it is clear from the context, m(I) will be simplified to m.

Let us define the coefficients ak(I),ck(I) for any I with maximal element m and kN through the following

Descent polynomial in “a-basis”

In this section, we investigate the coefficients ak(I). To do that, we will need to understand the coefficients of d(I,n) in the basis of nm+kk+1k=1m1, which is defined by the following equation d(I,n)=ā1(I)nm10+ā0(I)nm1++ām1(I)n1m.Observe that ā1(I)=0, since 0=d(I,m)=ā1(I)10+k=0m1āk(I)kk+1=ā1(I),therefore later on, we will concentrate on the coefficients āk(I) for 0km1. As it will turn out, all these coefficients are nonnegative integers, moreover, each of them

Descent polynomial in “c-basis”

The aim of the section is to give an affirmative answer for Conjecture 3.7 of [5], and to give some immediate consequences on the coefficients ck(I) and the evaluation of d(I,n) at negative integers. For corollaries considering the roots of d(I,n), see Section 5. We would like to remark that the proof of the positivity of the c-basis coefficients will be an algebraic manipulation rather than a combinatorial proof. However, giving such a proof could potentially imply a notion similar to

On the roots of d(I,n)

In this section, we will prove four propositions about the locations of the roots of d(I,n), two are for general I, and two are for some special cases. The first result is obtained by the technique of Theorem 4.16 of [5] based on the non-negativity of the coefficients ck(I). In the second result, we will prove a linear bound in m for the length of the roots of d(I,n), which will be based on the monotonicity of the coefficients āk(I). As a third result, we use similar arguments as in the proof

Some remarks and further directions

We described an interesting phenomenon in Section 4, namely that ck(I) and (1)md(I,n) are nonnegative integers. This result suggests that there might be some combinatorial proofs for them.

Question 6.1

What do the coefficients ck(I) and evaluations (1)md(I,n) count?

There are two conjectures about the roots of the descent polynomial that have been presented:

Proposition 6.2

Conjecture 4.3 of [5]

If z0 is a root of d(I,n), then

  • |z0|m,

  • z01.

This conjecture can be viewed as a special case of Theorem 5.10. As a common generalization of the two

CRediT authorship contribution statement

Ferenc Bencs: Conceptualized, Made the proofs, Wrote the manuscript.

Acknowledgments

I would like to express my sincere gratitude to Bruce Sagan, who pointed out some corollaries of the behavior of different coefficient sequences. I would also like to thank Alexander Diaz-Lopez for his helpful remarks. The research was partially supported by the MTA Rényi Institute Lendület Limits of Structures Research Group, Hungary. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme

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