Euler–Mahonian statistics and descent bases for semigroup algebras. (March 2018) - Braun, Benjamin; Olsen, McCabe
Abstract: We consider quotients of the unit cube semigroup algebra by particular Z r ≀ S n -invariant ideals. Using Gröbner basis methods, we show that the resulting graded quotient algebra has a basis where each element is indexed by colored permutations ( π, ϵ ) ∈ Z r ≀ S n and each element encodes the negative descent and negative major index statistics on ( π, ϵ ) . This gives an algebraic interpretation of these statistics that was previously unknown. This basis of the Z r ≀ S n -quotients allows us to recover certain combinatorial identities involving Euler–Mahonian distributions of statistics.