The Weak Bruhat Order and Separable Permutations

Details The Weak Bruhat Order and Separable Permutations The Weak Bruhat Order and Separable Permutations

2010-09-29

arxiv.org/abs/1009.5740v1

Mathematics

Combinatorics

Scientific paper

In this paper we consider the rank generating function of a separable permutation $\pi$ in the weak Bruhat order on the two intervals $[\text{id}, \pi]$ and $[\pi, w_0]$, where $w_0 = n,(n-1),..., 1$. We show a surprising result that the product of these two generating functions is the generating function for the symmetric group with the weak order. We then obtain explicit formulas for the rank generating functions on $[\text{id}, \pi]$ and $[\pi, w_0]$, which leads to the rank-symmetry and unimodality of the two graded posets.

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