William Schmitt

Incidence Hopf algebras

Abstract We present several results about incidence Hopf algebras of families of partially ordered sets, including a characterization of their algebra structure, a combinatorial technique for finding generating sets of primitive elements in the cocommutative case, and a determinant formula for the antipode which holds for a class including the Faa di Bruno Hopf algebra. We introduce a variety of examples of incidence Hopf algebras arising from families of graphs, matroids and distributive lattices, many of which generalize well-known Hopf algebras.

Linear trees and RNA secondary structure

The total number of RNA secondary structures of a given length with a fixed number of base pairs is computed, under the assumption that all base pairs can occur. This is done by establishing a one-to-one correspondence between secondary structures and trees. A duality operator on trees is presented, which explains a symmetry in the numbers counting secondary structures.

Antipodes and incidence coalgebras

Abstract We introduce the notion of a reduced incidence coalgebra of a family of locally finite partially ordered sets and describe how bialgebras and Hopf algebras arise as such coalgebras. In the case when a reduced incidence coalgebra is a Hopf algebra, we give explicit recursive and closed formulas for the antipode, each of which generalizes a corresponding formula from the theory of Mobius functions. We give applications to the inversion of ordinary formal power series and inversion of Dirichlet series. We also formulate the classical Lagrange inversion formula as a combinatorial formula for the antipode of a Hopf algebra arising from the family of finite partition lattices.

Multiple solutions of DNA restriction mapping problems

The construction of a restriction map of a DNA molecule from fragment length data is known to be NP hard. However, it is also known that under a simple model of randomness the number of solutions to the mapping problem increases exponentially with the length of the DNA molecule. In this paper we define a hierarchy of equivalence relations on the set of all solutions to the mapping problem and study the combinatorics and characterization of the equivalence classes.

Hopf algebras of combinatorial structures

A generalization of the definition of combinatorial species is given by considering functors whose domains are categories of finite sets, with various classes of relations as moronisms. Two cases in particular correspond to species for which one has notions of restriction and quotient of structures. Coalgebras and/or Hopf algebras can be associated to such species, the duals of which provide an algebraic framework for studying invariants of structures.

A free subalgebra of the algebra of matroids

This paper is an initial inquiry into the structure of the Hopf algebra of matroids with restriction-contraction coproduct. Using a family of matroids introduced by Crapo in 1965, we show that the subalgebra generated by a single point and a single loop in the dual of this Hopf algebra is free.

The free product of matroids

We introduce a noncommutative binary operation on matroids, called free product. We show that this operation respects matroid duality, and has the property that, given only the cardinalities, an ordered pair of matroids may be recovered, up to isomorphism, from its free product. We use these results to give a short proof of Welshs 1969 conjecture, which provides a progressive lower bound for the number of isomorphism classes of matroids on an n-element set.

A unique factorization theorem for matroids

We study the combinatorial, algebraic and geometric properties of the free product operation on matroids. After giving cryptomorphic definitions of free product in terms of independent sets, bases, circuits, closure, flats and rank function, we show that free product, which is a noncommutative operation, is associative and respects matroid duality. The free product of matroids M and N is maximal with respect to the weak order among matroids having M as a submatroid, with complementary contraction equal to N. Any minor of the free product of M and N is a free product of a repeated truncation of the corresponding minor of M with a repeated Higgs lift of the corresponding minor of N. We characterize, in terms of their cyclic flats, matroids that are irreducible with respect to free product, and prove that the factorization of a matroid into a free product of irreducibles is unique up to isomorphism. We use these results to determine, for K a field of characteristic zero, the structure of the minor coalgebra K{M} of a family of matroids M that is closed under formation of minors and free products: namely, K{M} is cofree, cogenerated by the set of irreducible matroids belonging to M.

Hopf algebra methods in graph theory

Abstract We study invariants of graphs, matroids and other combinatorial objects by considering certain associated Hopf algebras. We prove a structure theorem for these Hopf algebras which implies that any invariant which counts subobjects of a particular type is given by a unique polynomial in invariants which count connected subobjects. We introduce an invertible transformation between the set of all such invariants and the set of additive invariants.

An application of linear species

Abstract Combinatorial operations on linear species are used in order to obtain, in a simple manner, the identity 1 1−αx =1+ ∑ k⩾1 S(αk) x k 1−x k ′ where S (α; k ) denotes the number of aperiodic words of length k over an alphabet with α elements. The cyclotomic identity follows as an immediate corollary.