Generalized descent patterns in permutations and associated Hopf algebras

TL;DR

This paper introduces a new permutation statistic based on patterns of k consecutive letters, leading to the development of Hopf algebras that extend noncommutative symmetric and quasi-symmetric functions.

Contribution

It generalizes descent patterns in permutations to k-letter patterns and constructs associated Hopf algebras, broadening the algebraic framework of permutation statistics.

Findings

Defined a new permutation pattern statistic involving k consecutive letters

Constructed Hopf algebras generalizing noncommutative symmetric functions

Extended the algebraic structures to encompass new permutation patterns

Abstract

Descents in permutations or words are defined from the relative position of two consecutive letters. We investigate a statistic involving patterns of k consecutive letters, and show that it leads to Hopf algebras generalizing noncommutative symmetric functions and quasi-symmetric functions.