Line arrangements and r-Stirling partitions

A set partition of $[n] := {1, 2, dots, n }$ is called {em $r$-Stirling} if the numbers $1, 2, dots, r$ belong to distinct blocks. Haglund, Rhoades, and Shimozono constructed graded ring $R_{n,k}$ depending on two positive integers $k leq n$ whose algebraic properties are governed by the combinatorics of ordered set partitions of $[n]$ with $k$ blocks. We introduce a variant $R_{n,k}^{(r)}$ of this quotient for ordered $r$-Stirling partitions which depends on three integers $r leq k leq n$. We describe the standard monomial basis of $R_{n,k}^{(r)}$ and use the combinatorial notion of the {em coinversion code} of an ordered set partition to reprove and generalize some results of Haglund et. al. in a more direct way. Furthermore, we introduce a variety $X_{n,k}^{(r)}$ of line arrangements whose cohomology is presented as the integral form of $R_{n,k}^{(r)}$, generalizing results of Pawlowski and Rhoades.