Axel Hultman ; Svante Linusson ; John Shareshian ; Jonas Sjöstrand
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From Bruhat intervals to intersection lattices and a conjecture of Postnikov
dmtcs:3648 -
Discrete Mathematics & Theoretical Computer Science,
January 1, 2008,
DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)
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https://doi.org/10.46298/dmtcs.3648
From Bruhat intervals to intersection lattices and a conjecture of Postnikov
Authors: Axel Hultman 1; Svante Linusson 1; John Shareshian 2; Jonas Sjöstrand 3
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Axel Hultman;Svante Linusson;John Shareshian;Jonas Sjöstrand
1 Department of Mathematics [Sweden]
2 Department of Mathematics
3 Department of Mathematics and Physics
We prove the conjecture of A. Postnikov that ($\mathrm{A}$) the number of regions in the inversion hyperplane arrangement associated with a permutation $w \in \mathfrak{S}_n$ is at most the number of elements below $w$ in the Bruhat order, and ($\mathrm{B}$) that equality holds if and only if $w$ avoids the patterns $4231$, $35142$, $42513$ and $351624$. Furthermore, assertion ($\mathrm{A}$) is extended to all finite reflection groups.