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

Edward A. Bender, Gregory F. Lawler, Robin Pemantle +1

Let $n=b_1+...+b_k=b_1^{\prime }+\cdot +b_k^{\prime }$ 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…