In this paper, we construct several Hopf algebras of set systems with an automorphism group. The polynomial invariants of set systems studied in a previous paper are realized as Hopf algebra maps onto certain binomial and divided power Hopf algebras. An extended version of Stanley's symmetric function generalization of the chromatic polynomial is also realized as a Hopf algebra map. One of the main themes of this paper is that passage from a binomial to a divided power algebra corresponds, in the combinatorial setting, to the association of a group of automorphisms with a given set system. Several properties of binomial and divided power Hopf algebras have their origins in the combinatorial Hopf algebras; thus, we define delta operators, binomial sequences, divided power sequences, and investigate their properties. This paper is also relevant to the theory of formal groups and algebraic topology, by presenting a combinatorial proof of a familiar formal group law identity, and a combinatorial model for the covariant bialgebra of the universal formal group law.
In this paper, we investigate the umbral chromatic polynomial of a set system. This invariant was first defined for graphs by N. Ray and C. Wright [Ars Combin., 25B, 1988, 277-286], and it encodes the same information about the graph as R. Stanley's symmetric function generalization of the chromatic polynomial [Adv. Math., 111, 1995, p. 166]. We prove several identities for these two polynomials. Automorphism groups of set systems are also considered, and combinatorial interpretations and new formulas are given for the normalized versions of the polynomials.
In this preprint, we show that incidence Hopf algebras of partition lattices provide an efficient combinatorial framework for formal group theory and algebraic topology. We start by showing that the universal Hurwitz group law (respectively universal formal group law) are generating functions for certain trees. A formal group law identity with a combinatorial proof is also presented. We then illustrate the way in which several computations in algebraic topology can be carried out efficiently by using the incidence Hopf algebra framework; such computations include: expressing certain coactions, computing the images of the coefficients of the universal formal group law under the K-theory Hurewicz homomorphism, and proving certain congruences in the Lazard ring. We conclude by presenting two combinatorial models for the dual of the polynomial part of the modulo p Steenrod algebra, for a given prime p.