Computing the Size of Intervals in the Weak Bruhat Order

TL;DR

This paper studies the computational complexity of determining the size of intervals in the weak Bruhat order, providing polynomial-time algorithms for certain classes of permutations and analyzing the typical case complexity.

Contribution

It introduces polynomial-time algorithms for interval size computation when permutations have bounded width or intrinsic width, extending previous results, and analyzes average-case complexity for large permutations.

Findings

Interval size can be computed in polynomial time for permutations with bounded width.

Interval size computation is feasible in sub-exponential time for most large permutations.

The general problem remains open for arbitrary intervals.

Abstract

The weak Bruhat order on $ { \mathcal S }_n $ is the partial order $\prec$ so that $\sigma \prec \tau$ whenever the set of inversions of $\sigma$ is a subset of the set of inversions of $\tau$. We investigate the time complexity of computing the size of intervals with respect to $\prec$. Using relationships between two-dimensional posets and the weak Bruhat order, we show that the size of the interval $ [ \sigma_1, \sigma_2 ]$ can be computed in polynomial time whenever $\sigma_1^{-1} \sigma_2$ has bounded width (length of its longest decreasing subsequence) or bounded intrinsic width (maximum width of any non-monotone permutation in its block decomposition). Since permutations of intrinsic width $1$ are precisely the separable permutations, this greatly extends a result of Wei. Additionally, we show that, for large $n$, all but a vanishing fraction of permutations $ \sigma$ in $ { \mathcal S }_n$ give rise to intervals $ [ id , \sigma ]$ whose sizes can be computed with a sub-exponential time algorithm. The general question of the difficulty of computing the size of arbitrary intervals remains open.