Permutations with one or two 132-subsequences

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, of those containing exactly two such subsequences, yielding that {d,} is P-recursive. (~) 1998 Elsevier Science B.V.