Descent polynomials, peak polynomials and an involution on permutations

The size of the set of all permutations of n with a given descent set is a polynomial 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. We give a unitary expansion of descent polynomials in terms of peak polynomials. Then we use this expansion, along with an involution that flips the initial segment of a permutation, 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.