## Connecting descent and peak polynomials

Hacettepe Journal of Mathematics and Statistics, vol.53, no.2, pp.488-494, 2024 (SCI-Expanded)

• Publication Type: Article / Article
• Volume: 53 Issue: 2
• Publication Date: 2024
• Doi Number: 10.15672/hujms.1182500
• Journal Name:
• Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, zbMATH
• Page Numbers: pp.488-494
• Keywords: binomial coefficients, descent, peak, permutations
• Galatasaray University Affiliated: Yes

#### Abstract

A permutation σ = σ1 σ2 ... σn has a descent at i if σi > σi+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.