On Quasisymmetric Functions with Two Bordering Variables

We extend past results on a family of formal power series $K_{n,\mathrm{\Lambda }}$, parameterized by $n$ and $\mathrm{\Lambda }\subseteq [n]$, that largely resemble quasisymmetric functions. This family of functions was conjectured to have the property that the product $K_{n,\mathrm{\Lambda }}{K}_{m,\mathrm{\Omega }}$ of any two functions $K_{n,\mathrm{\Lambda }}$ and $K_{m,\mathrm{\Omega }}$ 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 $K_{n,\mathrm{\Lambda }}$'s forms an algebra. We also provide techniques for examining similar families of functions and a formula for the product $K_{n,\mathrm{\Lambda }}{K}_{m,\mathrm{\Omega }}$ when $n=1$.