Connecting descent and peak polynomials

Authors : Ezgi Kantarcı Oğuz

Pages : 488-494

Doi:10.15672/hujms.1182500

View : 144 | Download : 142

Publication Date : 2024-04-23

Article Type : Research Paper

Abstract :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.
Keywords : binomial coefficients, descent, peak, permutations