We give a monomial basis for *R*_{n,λ,s} in terms of (*n*,λmbda,*s*)-staircases, unifying the monomial bases studied by Garsia-Procesi and Haglund-Rhoades-Shimozono. We realize the
*S*_{n}-module structure of *R*_{n,λ,s} in terms of an action on (*n*,λ,*s*)-ordered set partitions. We find a formula for the Hilbert series of
*R*_{n,λ,s} in terms of inversion and diagonal inversion statistics on (*n*,λ,*s*)-ordered set partitions. Furthermore, we give an expansion of the graded Frobenius characteristic of our rings in terms of Gessel's fundamental basis and in terms of dual Hall-Littlewood symmetric functions.

We connect our work with Eisenbud-Saltman rank varieties using results of Weyman. As an application of our results on *R*_{n,λ,s}, we give a monomial basis, Hilbert series formula, and graded Frobenius characteristic formula for the coordinate ring of the scheme-theoretic intersection of a rank variety with diagonal matrices.

Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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