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