Line configurations and $r$-Stirling partitions
Line configurations and $r$-Stirling partitions
复制标题
生产线配置和 $r$-斯特林分区
DOI：
10.4310/joc.2019.v10.n3.a1
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发表时间：
2018
影响因子：
0.3
通讯作者：
A. Wilson
中科院分区：
文献类型：
--
作者：
B. Rhoades;A. Wilson
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.
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.