On the combinatorics of plethysm

We construct three (large, reduced) incidence algebras whose semigroups of multiplicative functions, under convolution, are anti-isomorphic, respectively, to the semigroups of what we call partitional, permutational and exponential formal power series without constant term, in infinitely many variables x = (x₁, x₂,…), under plethysm. We compute the Möbius function in each case. These three incidence algebras are the linear duals of incidence bialgebras arising, respectively, from the classes of transversals of partitions (with an order that we define), partitions compatible with permutations (with the usual refinement order), and linear transversals of linear partitions (with the order induced by that on transversals). We define notions of morphisms between partitions, permutations and linear partitions, respectively, whose kernels are defined to be, in each case, transversals, compatible partitions and linear transversals. We introduce, in each case, a pair of sequences of polynomials in x of binomial type, counting morphisms and monomorphisms, and obtain expressions for their connection constants, by summation and Möbius inversion over the corresponding posets of kernels.