Enumerations of maximum partial triple systems on 16 and 17 points

TL;DR

This paper completes the enumeration of maximum partial triple systems for small point sets, providing counts, structural properties, and automorphism groups for systems on 16 and 17 points, advancing combinatorial design classification.

Contribution

It provides the first complete enumeration and structural analysis of maximum partial triple systems on 16 and 17 points, including automorphism groups and configurations.

Findings

35810097 systems on 17 points identified

47744568 systems on 16 points identified

Structural properties and automorphism groups determined

Abstract

For $v\equiv 1$ or 3 (mod 6), maximum partial triple systems on $v$ points are Steiner triple systems, STS($v$)s. The 80 non-isomorphic STS(15)s were first enumerated around 100 years ago, but the next case for Steiner triple systems was unresolved until around 2004 when it was established that there are precisely 11084874829 non-isomorphic STS(19)s. In this paper we complete enumeration of non-isomorphic maximum partial triple systems for $v\le 19$. It is shown that there are 35810097 systems on 17 points and 47744568 on 16 points. We also establish that there are precisely 157151 non-isomorphic pairwise balanced designs, PBD($17,\{3,5\}$)s, having a single block of size 5. Structural properties of these systems are determined, including their automorphism groups, and the numbers of Pasch configurations, mitres and Fano planes contained in them. The systems themselves are available from the authors.