Inversion arrangements and the weak Bruhat order

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, Linusson, Shareshian and Sj\"{o}strand 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.