On the Stanley-Wilf Conjecture for the Number of Permutations Avoiding a Given Pattern

Abstract. Consider, for a permutation Sk, the number F(n, ) of permutations in Sn which avoid as a subpattern. The conjecture of Stanley and Wilf is that for every there is a constant c() < such that for all n, F(n, ) c()n . All the recent work on this problem also mentions the "stronger conjecture" that for every , the limit of F(n, )1/n exists and is finite. In this short note we prove that the two versions of the conjecture are equivalent, with a simple argument involving subadditivity. We also discuss n-permutations, containing all Sk as subpatterns. We prove that this can be achieved with n = k2 , we conjecture that asymptotically n (k/e)2 is the best achievable, and we present Noga Alon's conjecture that n (k/2)2 is the threshold for random permutations. Mathematics Subject Classification: 05A05,05A16.

• Type: Journal
• Year: 1999
• Where: COMBINATORICS
• Authors: Richard Arratia
• Keywords: COMBINATORICS 1999, Conjecture, Permutation Sk, Permutations