Permutations with one or two 132-subsequences

Abstract We prove a strikingly simple formula for the number of permutations containing exactly one subsequence of type 132. We show that this number equals the number of partitions of a convex ( n + 1)-gon into n − 2 parts by noncrossing diagonals. We also prove a recursive formula for the number d n of those containing exactly two such subsequences, yielding that { d n } is P -recursive.