A one-parameter deformation of the Farahat–Higman algebra

We show, by introducing an appropriate basis, that a one-parameter family of Hopf algebras introduced by Foissy [L. Foissy, Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations, Adv. Math. 218 (1) (2008) 136–162] interpolates between the Faà di Bruno algebra and the Farahat–Higman algebra. Its structure constants in this basis are deformations of the top connection coefficients, for which we obtain analogues of Macdonald’s formulas.