Inversion polynomials for permutations avoiding consecutive patterns

Abstract

In 2012, Sagan and Savage introduced the notion of st-Wilf equivalence for a statistic st and for sets of permutations that avoid particular permutation patterns. In this paper we consider inv-Wilf equivalence on sets of permutations that avoid two or more consecutive permutation patterns. We say that two sets of generalized permutation patterns Π and Π′ are inv-Wilf equivalent if the generating function for the inversion statistic on the permutations that simultaneously avoid all elements of Π is equal to the generating function for the inversion statistic on the permutations that simultaneously avoid all elements of Π′.In 2013, Cameron and Killpatrick gave the inversion generating function for Fibonacci tableaux which are in one-to-one correspondence with the set of permutations that simultaneously avoid the consecutive patterns 321 and 312. In this paper, we use the language of Fibonacci tableaux to study the inversion generating functions for permutations that avoid Π where Π is a set of three or more consecutive permutation patterns. In addition, we introduce the more general notion of strip tableaux which are a useful combinatorial object for studying consecutive pattern avoidance. We give the inversion generating functions for Π a subset of 4 or 5 consecutive permutation patterns and for all but one of the cases where Π is a subset of three consecutive permutation patterns.