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From Bruhat intervals to intersection lattices and a conjecture of Postnikov

Abstract : 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.
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Axel Hultman, Svante Linusson, John Shareshian, Jonas Sjöstrand. From Bruhat intervals to intersection lattices and a conjecture of Postnikov. 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), 2008, Viña del Mar, Chile. pp.203-214. ⟨hal-01185184⟩

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