Usefulness of recursive subdivision via blocking ideals

Determine whether the divide-and-conquer decomposition of the number of linear extensions \(e(P)\), obtained by splitting at pairs of incomparable elements and recursively computing the contributions of blocking ideals, is useful for some classes of finite posets, and determine whether the computational cost of calculating the blocking ideals outweighs the gains from the recursive subdivision.

Background

The paper derives a decomposition of the total number of linear extensions of a finite poset into sums involving blocking ideals and the numbers of linear extensions of smaller induced subposets. This suggests a possible divide-and-conquer algorithm for computing $e(P)$. The paper explicitly leaves unresolved whether this strategy provides practical or theoretical benefits for any classes of posets, given the cost of generating the blocking ideals.