Some coefficient sequences related to the descent polynomial
Introduction
Denote the group of permutations on by . For a permutation , the set of descending positions is We would like to investigate the number of permutations in with a fixed descent set. More precisely, for a finite , let . Then for , we can count the number of permutations with descent set that we will denote by
In 1915, this function was shown to be a degree polynomial in 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 the set of indexes are the peak positions. They have proved that the counting function is a product of a polynomial and the exponential . Also in the same paper, they proposed some conjectures about the coefficients of 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 is defined uniquely through the following equation: In [5], it was shown that the sequence 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 , then the sequence is log-concave, that means that for any we have
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 :
Theorem 5.3 If and for some , then .
As in [5] we will also consider the coefficient sequence, that is defined by the following equation By using a new recursion from [5] we prove that
Proposition 4.3 If , then for any the coefficient .
In the last section, we will establish zero-free regions for descent polynomials. In particular, we will prove the following.
Theorem 5.10 If and for some , then . In particular, , where is the real part of .
This paper is organized as follows. In the next section we will define two sequences, and , 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 . In Section 4, we will prove a conjecture concerning the sequence and some consequences. In Section 5, we prove some bounds on the roots of the descent polynomials that will rely on a result of and some further investigation of the sequence .
Section snippets
Preliminaries
In this section, we will recall some recursions of the descent polynomial associated with a set 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 by , the maximal element of by . If it is clear from the context, will be simplified to .
Let us define the coefficients for any with maximal element and through the following
Descent polynomial in “-basis”
In this section, we investigate the coefficients . To do that, we will need to understand the coefficients of in the basis of , which is defined by the following equation Observe that , since therefore later on, we will concentrate on the coefficients for . As it will turn out, all these coefficients are nonnegative integers, moreover, each of them
Descent polynomial in “-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 and the evaluation of at negative integers. For corollaries considering the roots of , see Section 5. We would like to remark that the proof of the positivity of the -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
In this section, we will prove four propositions about the locations of the roots of , two are for general , 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 . In the second result, we will prove a linear bound in for the length of the roots of , which will be based on the monotonicity of the coefficients . 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 and are nonnegative integers. This result suggests that there might be some combinatorial proofs for them.
Question 6.1 What do the coefficients and evaluations count?
There are two conjectures about the roots of the descent polynomial that have been presented:
Proposition 6.2 If is a root of , then , .Conjecture 4.3 of [5]
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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