On Quasisymmetric Functions with Two Bordering Variables

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

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