Zimmermann Type Cancellation in the Free Faa di Bruno Algebra

Abstract

The N-variable Hopf algebra introduced by Brouder, Fabretti, and Krattenaler (BFK) in the context of non-commutative Lagrange inversion can be identified with the inverse of the incidence algebra of N-colored interval partitions. The (BFK) antipode and its reflection determine the (generally distinct) left and right inverses of power series with non-commuting coefficients and N non-commuting variables. As in the case of the Faa di Bruno Hopf algebra, there is an analogue of the Zimmermann cancellation formula. The summands of the (BFK) antipode can indexed by the depth first ordering of vertices on contracted planar trees, and the same applies to the interval partition antipode. Both can also be indexed by the breadth first ordering of vertices in the non-order contractible planar trees in which precisely one non-degenerate vertex occurs on each level.