Geodesic
From Bruhat intervals to intersection lattices and a conjecture of Postnikov
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) (2008).

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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. 
@article{DMTCS_2008_special_255_a56,
     author = {Hultman, Axel and Linusson, Svante and Shareshian, John and Sj\"ostrand, Jonas},
     title = {From {Bruhat} intervals to intersection lattices and a conjecture of {Postnikov}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)},
     year = {2008},
     doi = {10.46298/dmtcs.3648},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3648/}
}
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%A Sjöstrand, Jonas
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%D 2008
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Hultman, Axel; Linusson, Svante; Shareshian, John; Sjöstrand, Jonas. From Bruhat intervals to intersection lattices and a conjecture of Postnikov. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) (2008). doi : 10.46298/dmtcs.3648. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3648/

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