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

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 '-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.
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Taxonomy
TopicsStochastic processes and statistical mechanics · Bayesian Methods and Mixture Models · Data Management and Algorithms
