# Irreducible Compositions and the First Return to the Origin of a Random Walk

Abstract. Let n = b1 + ... + bk = b1' + ... + bk' be a pair of compositions of n into k positive parts. We say this pair is {\em irreducible} if there is no positive j<k for which b1 + ... + bj = b1' + ... + bj'. The probability that a random pair of compositions of n is irreducible is shown to be asymptotic to 8/n. This problem leads to a problem in probability theory. Two players move along a game board by rolling a die, and we ask when the two players will first coincide. A natural extension is to show that the probability of a first return to the origin at time n for any mean-zero variance V random walk is asymptotic to \sqrt{V/(2\pi)} n-3/2. We prove this via two methods, one analytic and one probabilistic.

Received: April 13, 2004. Accepted: September 1, 2004. Final Version: September 19, 2004.

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