Computing the Size of Intervals in the Weak Bruhat Order

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.