Abstract
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 μ k = E(ξk) < 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.
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Szabados, T., Székely, B. Moments of an exponential functional of random walks and permutations with given descent sets. Periodica Mathematica Hungarica 49, 131–139 (2004). https://doi.org/10.1023/B:MAHU.0000040544.59987.08
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DOI: https://doi.org/10.1023/B:MAHU.0000040544.59987.08