{\Gamma}-species and the enumeration of k-trees

TL;DR

This paper develops a new combinatorial framework using $\Gamma$-species to efficiently enumerate unlabeled k-trees, providing recursive formulas, computational tables, and Sage code for practical calculations.

Contribution

It introduces a novel application of $\Gamma$-species to derive recursive equations for counting unlabeled k-trees, enabling fast enumeration up to certain sizes.

Findings

Derived recursive functional equations for k-trees enumeration.

Provided enumeration tables for k up to 10 and n up to 30.

Included Sage code for practical computation of k-trees counts.

Abstract

We study the class of graphs known as k-trees through the lens of Joyal's theory of combinatorial species (and an equivariant extension known as '$\Gamma$-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, efficient 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.