G{cyrillic}-species and the enumeration of k-trees

Abstract

We study the class of graphs known as k-trees through the lens of Joyal's theory of combinatorial species (and an extension known as -species' which incorporates data about 'structural' group actions). This culminates in a system of recursive functional equations giving the generating function for unlabeled k-trees which allows for fast, effcient computation of their numbers. Enumerations up to k = 10 and n = 30 (for a k-tree with n + k-1 vertices) are included in tables, and Sage code for the general computation is included in an appendix.