Constraining strong c-Wilf equivalence using cluster poset asymptotics

Abstract

Let π∈Sm and σ∈Sn be permutations. An occurrence of π in σ as a consecutive pattern is a subsequence σiσi+1⋯σi+m−1 of σ with the same order relations as π. We say that patterns π,τ∈Sm are strongly c-Wilf equivalent if for all n and k, the number of permutations in Sn with exactly k occurrences of π as a consecutive pattern is the same as for τ. In 2018, Dwyer and Elizalde [6] conjectured (generalizing a conjecture of Elizalde [8] from 2012) that if π,τ∈Sm are strongly c-Wilf equivalent, then (τ1,τm) is equal to one of (π1,πm), (πm,π1), (m+1−π1,m+1−πm), or (m+1−πm,m+1−π1). We prove this conjecture using the cluster method introduced by Goulden and Jackson in 1979 [12], which Dwyer and Elizalde used to prove that |π1−πm|=|τ1−τm|. A consequence of our result is the full classification of c-Wilf equivalence for a special class of permutations, the non-overlapping permutations. Our approach uses analytic methods to approximate the number of linear extensions of the “cluster posets” defined by Elizalde and Noy in 2012 [11].