Irreducible compositions and the first return to the origin of a random walk

Let n = b1 + � � � + bk = b ' + � � � + bbe a pair of compositions of n into k positive parts. We say this pair is irreducible if there is no positive j < k for which b1 + � � � + bj = b ' + � � � + b ' . 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 p V/(2�)n −3/2 . We prove this via two methods, one