Hopf Algebra on Vincular Permutation Patterns

TL;DR

This paper introduces a novel Hopf algebra structure on pairs of interval partitions and permutations to model vincular permutation patterns, facilitating algebraic analysis of pattern occurrences.

Contribution

It presents the first Hopf algebra framework that captures vincular patterns through pairs of interval partitions and permutations.

Findings

The algebra encodes vincular pattern occurrences algebraically.

Linear functionals counting pattern occurrences are compatible with the Hopf algebra operations.

Provides a new algebraic tool for studying vincular permutation patterns.

Abstract

We introduce a new Hopf algebra that operates on pairs of finite interval partitions and permutations of equal length. This algebra captures vincular patterns, which involve specifying both the permutation patterns and the consecutive occurrence of values. Our motivation stems from linear functionals that encode the number of occurrences of these patterns, and we show that they behave well with respect to the operations of this Hopf algebra.