Inclusion-exclusion, partial representations of semigroups, and nonassociative Specht polynomials

The dimension of the space of multilinear products of higher commutators is equal to the number of derangements, $[e^{-1}n!]$. Our search for a combinatorial explanation for this fact led us to study representations of left regular bands, whose resolution is obtained through analysis of cubical partial representations. There are applications in combinatorics, probability, and nonassociative algebra.