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

Bounded affine permutations I. Pattern avoidance and enumeration

We introduce a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We study pattern avoidance in bounded affine permutations. In particular, we show that if $\tau$ is one of the finite increasing oscillations, then every $\tau$-avoiding affine permutation satisfies the boundedness condition. We also explore the enumeration of pattern-avoiding affine permutations that can be decomposed into blocks, using analytic methods to relate their exact and asymptotic enumeration to that of the underlying ordinary permutations. Finally, we perform exact and asymptotic enumeration of the set of all bounded affine permutations of size $n$. A companion paper will focus on avoidance of monotone decreasing patterns in bounded affine permutations.Comment: 35 page