Using homological duality in consecutive pattern avoidance

TL;DR

This paper introduces a homological duality approach to identify when different collections of permutation patterns share the same exponential generating functions for pattern avoidance, using a length-preserving bijection and an efficient inverse generating function algorithm.

Contribution

It provides a sufficient condition based on pattern bijections for equal generating functions and develops a fast algorithm for computing inverse generating functions, including differential equations.

Findings

Identifies conditions for equal exponential generating functions in pattern avoidance.

Develops a direct algorithm for inverse generating function computation.

Derives linear differential equations for inverse generating functions.

Abstract

Using the approach suggested in [arXiv:1002.2761] we present below a sufficient condition guaranteeing that two collections of patterns of permutations have the same exponential generating functions for the number of permutations avoiding elements of these collections as consecutive patterns. In short, the coincidence of the latter generating functions is guaranteed by a length-preserving bijection of patterns in these collections which is identical on the overlappings of pairs of patterns where the overlappings are considered as unordered sets. Our proof is based on a direct algorithm for the computation of the inverse generating functions. As an application we present a large class of patterns where this algorithm is fast and, in particular, allows to obtain a linear ordinary differential equation with polynomial coefficients satisfied by the inverse generating function.