On Quasisymmetric Functions with Two Bordering Variables

TL;DR

This paper proves that a family of formal power series related to quasisymmetric functions forms an algebra under multiplication, extending previous results and providing formulas for specific cases.

Contribution

It establishes that the family of functions $K_{n, Lambda}$ forms an algebra and introduces techniques for analyzing similar families.

Findings

The span of $K_{n, Lambda}$ functions is closed under multiplication.

A formula for the product $K_{n, Lambda}K_{m, Omega}$ when $n=1$ is provided.

The family of functions forms an algebra, confirming a conjecture.

Abstract

We extend past results on a family of formal power series $K_{n, \Lambda}$, parameterized by $n$ and $\Lambda \subseteq [n]$, that largely resemble quasisymmetric functions. This family of functions was conjectured to have the property that the product $K_{n, \Lambda}K_{m, \Omega}$ of any two functions $K_{n, \Lambda}$ and $K_{m, \Omega}$ from the family can be expressed as a linear combination of other functions from the family. In this paper, we show that this is indeed the case and that the span of the $K_{n, \Lambda}$'s forms an algebra. We also provide techniques for examining similar families of functions and a formula for the product $K_{n, \Lambda}K_{m, \Omega}$ when $n=1$.