Permutation Designs and Sequencing Highly Transitive Group Actions

TL;DR

This paper explores permutation design problems using sharply transitive group actions, providing explicit constructions for prime-sized sets and analyzing algorithms for complex cases, including designs based on Mathieu groups.

Contribution

It introduces a new approach to permutation design using sharply transitive groups, with explicit constructions and analysis of algorithms for complex cases.

Findings

Explicit construction for prime-sized sets with t=3

Branching algorithm for general cases, including t=6 with Mathieu group M12

Most sharply transitive group actions lead to solutions, with some exceptions

Abstract

We consider an experimental design problem for permutations: given a fixed set $X$, and an integer $t$, construct a list $L$ of permutations of $X$ such that every ordered $t$-tuple of distinct elements of $X$ occurs as a consecutive subsequence of exactly one permutation in $L$. In this paper we focus on solutions based on sharply transitive group actions, in effect generalizing Gordon's notion of group sequencing. We give an explicit construction when $|X|$ is prime for the case $t=3$, and analyze a branching algorithm for the general case which produces, for example, a rare design with $t=6$ based on the Mathieu group $M_{12}$, and suggests that every sharply transitive group action leads to a solution, apart from an explicit list of counterexamples. We state a number of conjectures and indicate directions for future work.