Computational Approaches to Consecutive Pattern Avoidance in Permutations

Abstract

In recent years, there has been increasing interest in consecutive pattern avoidance in permutations. In this paper, we introduce two approaches to counting permutations that avoid a set of prescribed patterns consecutively. These algoritms have been implemented in the accompanying Maple package CAV, which can be downloaded from the author's website. As a byproduct of the first algorithm, we have a theorem giving a sufficient condition for when two pattern sets are strongly (consecutively) Wilf-Equivalent. For the implementation of the second algorithm, we define the cluster tail generating function and show that it always satisfies a certain functional equation. We also explain how the CAV package can be used to approximate asymptotic constants for single pattern avoidance.

The most and the least avoided consecutive patterns

“…21) is the one for which the number of permutations avoiding it is asymptotically largest. This conjecture is often mentioned in the literature [4,5,6,18]. The first main result of the present paper is a proof of this conjecture.…”

“…We say that two patterns σ and τ are strongly c-Wilf-equivalent if P σ (u, z) = P τ (u, z), and that they are c-Wilf-equivalent if P σ (0, z) = P τ (0, z). The last condition can be rephrased as α n (σ) = α n (τ ) for all n. Nakamura [18,Conjecture 6] conjectures that two patterns are strongly c-Wilf-equivalent iff they are c-Wilf-equivalent. A complete classification into c-Wilf-equivalence classes is known for patterns of length up to 6, and in these cases they coincide with strong c-Wilf-equivalence classes.…”

Shuffle algebras, homology, and consecutive pattern avoidance

“…Unlike that, our approach gives formulas free from those trivial cancellations. Further progress in algorithmic and computational approaches to consecutive pattern avoidance is presented in recent preprints [2,29]. We also wish to mention a follow-up [18] to an earlier version of this paper showing the relevance of homological methods for studying consecituve patterns.…”

Pattern avoidance in labelled trees

“…As in the case of permutations, we expect that at least in the case of a single pattern a careful study of its self-overlaps ("overlap sets" of [23], or equivalently "overlap maps" of [31]) would be very beneficial for studying Wilf equivalence. We shall discuss it in detail elsewhere, mentioning a particular case briefly in the next section.…”

“…. n (the identity permutation) is the easiest to avoid, and to the conjecture of Nakamura [31] that the permutation 12 . .…”