范久瑜^*

（  四川大学数学学院，成都　610064；  ）

摘要： 对每个排列$w$,我们可以根据$w$的逆序构造一个超平面排列$mathcal{A}_w$,称为$w$对应的逆序超平面排列。$mathcal{A}_w$中区域的个数小于等于在（强）布吕阿序下比$w$小的排列个数,取等号当且仅当$w$ 避免模式4231, 35142, 42513 和 351624。 这个结果最先是由 Postnikov 提出猜想,后来被Hultman等人证明的。在本文中,我们证明了$mathcal{A}_w$中区域的个数大于等于在弱布吕阿序下比$w$小的排列个数,取等号当且仅当$w$避免模式231 和 312。
关键词： 逆序超平面排列,弱布吕阿序,模式避免

Neil J.Y. Fan^*

（  Department of Mathematics, Sichuan University, Chengdu 610064；  ）

Abstract： For each permutation $w$, we can construct a collection of hyperplanes $mathcal{A}_w$ according to the inversions of $w$, which is called the inversion hyperplane arrangement associated to $w$. It was conjectured by Postnikov and confirmed by Hultman et al. that the number of regions of $mathcal{A}_w$ is less than or equal to the number of permutations below $w$ in the Bruhat order, with the equality holds if and only if $w$ avoids the four patterns 4231, 35142, 42513 and 351624. In this paper, we show that the number of regions of $mathcal{A}_w$ is greater than or equal to the number of permutations below $w$ in the weak Bruhat order, with the equality holds if and only if $w$ avoids the patterns 231 and 312.

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