On the Distribution of the Number of Occurrences of an Order-Preserving Pattern of Length Three in a Random Permutation

Abstract

Recently, a considerable number of papers in computer science and mathematics examined the number of permutations containing exactly s occurrences of a prescribed order-preserving pattern (or forbidden pattern). It is well known that, mathematically, this is an NP-hard problem. Even in the simple case where the length of an order-preserving pattern is three, the number of permutations of size n containing s (s ≥ 3) order-preserving patterns remains unknown (see Fulmek Adv Appl Math 30(4):607–632, 2003). This manuscript provides a probabilistic approach to enumerate the number of permutations that contain exactly s occurrences of an order-preserving pattern of length three. The method is based on the insertion procedure of the finite Markov chain imbedding technique. Numerical results are provided to illustrate the theoretical results.