Abstract. A permutation σ = σ1σ2 · · · 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 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. 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.