Connecting descent and peak polynomials
Ezgi Kantarci Oğuz
TL;DR
The paper addresses the relationship between descent polynomials $d(S,n)$ and peak polynomials $p(I,n)$ for permutations, introducing a unitary expansion of $d(S,n)$ in terms of $p(I,n)$ and extending the framework to marked permutations. It develops a combinatorial mechanism based on flip involutions to partition descent-sets and to obtain an inclusion-exclusion expression for peak coefficients, enabling a binomial-basis interpretation of these coefficients and providing a new proof of peak-polynomial positivity. A key contribution is the construction of a concrete, interpretable link between descent and peak statistics, via marked permutations and involutions, culminating in explicit combinatorial formulas for the peak-coefficients. The results advance understanding of the structure of permutation statistics and their positivity properties, with potential implications for related combinatorial polynomials and their bases.
Abstract
A permutation $σ=σ_1 σ_2 \cdots σ_n$ has a descent at $i$ if $σ_i>σ_{i+1}$. 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 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 a combinatorial interpretation of the coefficients of the peak polynomial in a binomial basis, thus giving a new proof of the peak polynomial positivity conjecture.