On Quasisymmetric Functions with Two Bordering Variables

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 $n=1$.

16 pages, 0 figures