4 research outputs found
Combinatorial species and graph enumeration
In enumerative combinatorics, it is often a goal to enumerate both labeled
and unlabeled structures of a given type. The theory of combinatorial species
is a novel toolset which provides a rigorous foundation for dealing with the
distinction between labeled and unlabeled structures. The cycle index series of
a species encodes the labeled and unlabeled enumerative data of that species.
Moreover, by using species operations, we are able to solve for the cycle index
series of one species in terms of other, known cycle indices of other species.
Section 3 is an exposition of species theory and Section 4 is an enumeration of
point-determining bipartite graphs using this toolset. In Section 5, we extend
a result about point-determining graphs to a similar result for
point-determining {\Phi}-graphs, where {\Phi} is a class of graphs with certain
properties. Finally, Appendix A is an expository on species computation using
the software Sage [9] and Appendix B uses Sage to calculate the cycle index
series of point-determining bipartite graphs.Comment: 39 pages, 16 figures, senior comprehensive project at Carleton
Colleg
The explicit molecular expansion of the combinatorial logarithm
Just as the power series of is the analytical substitutional inverse of the series of , the (virtual) combinatorial species, , is the combinatorial substitutional inverse of the combinatorial species, , of non-empty finite sets. This , , has been introduced by A. Joyal in 1986 by making use of an iterative scheme. Given a species (with ), one of its main applications is to express the species, , of -structures through the formula where denotes the species of non-empty -structures. Since its creation, equivalent descriptions of the combinatorial logarithm have been given by other combinatorialists (G. L., I. Gessel, J. Li), but its exact decomposition into irreducible components (molecular expansion) remained unclear. The main goal of the present work is to fill this gap by computing explicitly the molecular expansion of the combinatorial logarithm and of , a "cousin'' of the tensorial species, , of free Lie algebras