The algebraic combinatorics of snakes

TL;DR

This paper explores the algebraic combinatorics of snakes, a generalization of alternating permutations in Coxeter groups, using noncommutative symmetric functions to reveal algebraic identities and properties.

Contribution

It introduces a framework connecting snakes with combinatorial Hopf algebras, extending properties of generating functions to algebraic identities in noncommutative symmetric functions.

Findings

Algebraic identities for type B noncommutative symmetric functions

Expressions of generating functions as trigonometric functions

Lifting properties of snakes to algebraic structures

Abstract

Snakes are analogues of alternating permutations defined for any Coxeter group. We study these objects from the point of view of combinatorial Hopf algebras, such as noncommutative symmetric functions and their generalizations. The main purpose is to show that several properties of the generating functions of snakes, such as differential equations or closed form as trigonometric functions, can be lifted at the level of noncommutative symmetric functions or free quasi-symmetric functions. The results take the form of algebraic identities for type B noncommutative symmetric functions, noncommutative supersymmetric functions and colored free quasi-symmetric functions.