Line configurations and $r$-Stirling partitions

A set partition of $[n] := \lbrace 1, 2, \dotsc, n \rbrace$ is called $r$-Stirling if the numbers $1, 2, \dotsc, r$ belong to distinct blocks. Haglund, Rhoades, and Shimozono constructed a 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 $R^{(r)}_{n,k}$ 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 $R^{(r)}_{n,k}$ and use the combinatorial notion of the 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 $X^{(r)}_{n,k}$ of line configurations whose cohomology is presented as the integral form of $R^{(r)}_{n,k}$, generalizing results of Pawlowski and Rhoades.

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B. Rhoades was partially supported by NSF Grant DMS-1500838. A. T. Wilson was partially supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship. The authors thank an anonymous referee for their careful reading of the paper and, in particular, for pointing out the applicability of the Leray-Hirsch Theorem. We thank Jeff Remmel for his mathematics, mentoring, and friendship.

Received 21 April 2018

Published 23 May 2019