  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.
