Asymmetric Latin squares, Steiner triple systems, and edge-parallelisms

TL;DR

This paper discusses the asymmetry of Latin squares, Steiner triple systems, and edge-parallelisms, highlighting that almost all such objects are asymmetric, with references to prior results and historical context.

Contribution

It provides a historical account and clarification of the asymmetry results for these combinatorial objects, emphasizing that most are asymmetric.

Findings

Almost all Latin squares are asymmetric.

Almost all Steiner triple systems are asymmetric.

The paper references established asymmetry results for these objects.

Abstract

This article, showing that almost all objects in the title are asymmetric, is re-typed from a manuscript I wrote somewhere around 1980 (after the papers of Bang and Friedland on the permanent conjecture but before those of Egorychev and Falikman). I am not sure of the exact date. The manuscript had been lost, but surfaced among my papers recently. I am grateful to Laci Babai and Ian Wanless who have encouraged me to make this document public, and to Ian for spotting a couple of typos. In the section on Latin squares, Ian objects to my use of the term "cell"; this might be more reasonably called a "triple" (since it specifies a row, column and symbol), but I have decided to keep the terminology I originally used. The result for Latin squares is in B. D. McKay and I. M. Wanless, On the number of Latin squares, Annals of Combinatorics 9 (2005), 335-344 (arXiv 0909.2101), while the result for Steiner triple systems is in L. Babai, Almost all Steiner triple systems are asymmetric, Annals of Discrete Mathematics 7 (1980), 37-39.