Shuffle algebras, homology, and consecutive pattern avoidance

TL;DR

This paper explores the algebraic structure of shuffle algebras and applies homological methods to derive new results on pattern avoidance in permutations, including exact counts and asymptotic behavior.

Contribution

It introduces homological techniques to analyze shuffle algebras with monomial relations and applies these to solve problems in permutation pattern avoidance.

Findings

Homological results on shuffle algebras with monomial relations

Exact enumeration results for consecutive pattern avoidance

Asymptotic analysis of pattern avoidance in permutations

Abstract

Shuffle algebras are monoids for an unconvential monoidal category structure on graded vector spaces. We present two homological results on shuffle algebras with monomial relations, and use them to prove exact and asymptotic results on consecutive pattern avoidance in permutations.