Pattern avoidance in ordered set partitions and words

Abstract

We consider the enumeration of ordered set partitions avoiding a permutation pattern, as introduced by Godbole, Goyt, Herdan and Pudwell. Let opn,k(p) be the number of ordered partitions of {1,2,…,n} into k blocks that avoid a permutation pattern p. We establish an explicit identity between the number opn,k(p) and the number of words avoiding the inverse of p. This identity allows us to easily translate results on pattern-avoiding words obtained in earlier works into equivalent results on pattern-avoiding ordered set partitions. In particular, (a) we determine the asymptotic growth rate of the sequence (opn,k(p))n≥1 for every positive k and every permutation pattern p, (b) we partially confirm a conjecture of Godbole et al. concerning the variation of the sequences (opn,k(p))1≤k≤n, (c) we undertake a detailed study of the number of ordered set partitions avoiding a permutation pattern of length 3. By the way, we observe that the number of words on k letters of length n avoiding a permutation pattern of length 3 can be written as a single sum by simplifying a formula of Burstein.