The Classification Of 231-Avoiding Permutations By Descents And Maximum Drop

We study the number of 231-avoiding permutations of length n with j-descents and maximum drop of less than or equal to k, which we denote by a(n,j)((k)). We show that a(n,j)((k)) also counts the number of Dyck paths of length 2n with n - j peaks and height <= k + 1, and the number of ordered trees with n edges, j + 1 internal nodes, and height <= k + 1. We show that the generating functions for the a(n,j) ((k)) s with k fixed satisfy a simple recursion. We also use the combinatorics of ordered trees to prove new explicit formulas for a(n,j)((k)) and prove a simple recursion for the a(n,j)((k))s.