Line configurations and r-Stirling partitions

TL;DR

This paper introduces a new algebraic structure and geometric variety related to r-Stirling partitions, generalizing previous work and providing explicit bases and cohomology descriptions for these objects.

Contribution

It defines a variant of the graded ring for r-Stirling partitions, describes its basis via coinversion codes, and links it to a geometric variety of line arrangements.

Findings

Established the standard monomial basis for R_{n,k}^{(r)}.

Reproved and extended results on ordered set partitions.

Connected the algebraic structure to the cohomology of a line arrangement variety.

Abstract

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.