Moments of an exponential functional of random walks and permutations with given descent sets

The exponential functional of simple, symmetric random walks with negative drift is an infinite polynomial Y = 1+ ξ1 + ξ1ξ2 + ξ1ξ2ξ3 + · · · of independent and identically distributed non-negative random variables. It has moments that are rational functions of the variables μk = E(ξ ) < 1 with universal coefficients. It turns out that such a coefficient is equal to the number of permutations with descent set defined by the multiindex of the coefficient. A recursion enumerates all numbers of permutations with given descent sets in the form of a Pascal-type triangle.