[Paper Review] Cycle structures of autotopisms of the Latin squares of order up to 11

Raúl Manuel Falcón Ganfornina|arXiv (Cornell University)|Sep 19, 2007

graph theory and CDMA systems6 references20 citations

TL;DR

This paper classifies all possible cycle structures of autotopisms in Latin squares of order up to 11, establishing that the number of Latin squares admitting a given isotopism depends solely on the cycle structure of that isotopism. The authors derive theoretical constraints on cycle structures using permutation cycle decomposition and fixed-point analysis, then implement a computational algorithm to enumerate all valid autotopism cycle types, providing a complete catalog for orders 1 through 11.

ABSTRACT

The cycle structure of a Latin square autotopism $Θ=(α,β,γ)$ is the triple $(\mathbf{l}_α,\mathbf{l}_β,\mathbf{l}_γ)$, where $\mathbf{l}_δ$ is the cycle structure of $δ$, for all $δ\in\{α,β,γ\}$. In this paper we study some properties of these cycle structures and, as a consequence, we give a classification of all autotopisms of the Latin squares of order up to 11.

Motivation & Objective

• To classify all possible cycle structures of autotopisms in Latin squares of order up to 11.
• To determine which cycle structures of isotopisms can actually occur as autotopisms of some Latin square.
• To establish that the number of Latin squares admitting a given isotopism depends only on the cycle structure of the isotopism.
• To provide a computational catalog of all such valid cycle structures for orders 1 to 11.

Proposed method

• The cycle structure of a permutation is defined as the tuple (l₁, l₂, ..., lₙ), where lᵢ is the number of cycles of length i.
• The paper uses the classification of non-trivial autotopisms from McKay, Meynert, and Myrvold, which divides autotopisms into three cases based on fixed points and cycle structure symmetry.
• For each isotopism Θ = (α, β, γ), the cycle structure (l_α, l_β, l_γ) is computed and validated against necessary conditions derived from quasigroup isotopy and Latin square orthogonality.
• A computer program is implemented to systematically generate and filter all possible cycle structure triples that satisfy the theoretical constraints and correspond to actual autotopisms.
• The algorithm checks whether a given cycle structure triple can yield a Latin square via the orthogonal array representation and isotopism invariance.
• The method verifies that only cycle structures consistent with Theorem 1 and the fixed-point constraints are retained, excluding invalid configurations like (0,0,1,0,0,1,0,0,0,0) for n=6.

Experimental results

Research questions

• RQ1 Which cycle structure triples (l_α, l_β, l_γ) can arise from autotopisms of Latin squares of order up to 11?
• RQ2 How do fixed points and cycle type symmetry constrain the possible autotopism structures in Latin squares?
• RQ3 Can the number of Latin squares admitting a given isotopism Θ be determined solely from the cycle structure of Θ?
• RQ4 What is the complete set of cycle structures that occur as autotopisms in Latin squares of order 1 through 11?
• RQ5 Which cycle structure triples are theoretically possible but do not correspond to any actual Latin square autotopism?

Key findings

• The paper provides a complete catalog of all cycle structures of autotopisms for Latin squares of order 1 to 11, with Tables 4–6 listing all valid (l_α, l_β, l_γ) triples.
• For order 11, the only cycle structure with a single 11-cycle in all components is (0,0,0,0,0,0,0,0,0,0,1), and this occurs in the autotopism group of at least one Latin square.
• The cycle structure (3,4,0,0,0,0,0,0,0,0,0) for all three permutations is valid and corresponds to an autotopism of a Latin square of order 11.
• The cycle structure (0,5,0,0,0,0,0,0,0,0,0) for all three permutations is invalid for n=11, as it does not correspond to any actual autotopism.
• For n=10, the cycle structure (0,0,0,0,0,0,0,0,0,1) in all components is valid and corresponds to a Latin square with a single 10-cycle in each permutation.
• The paper identifies and excludes several theoretically plausible cycle structures—such as (0,0,1,0,0,1,0,0,0) for n=6—that do not correspond to any actual autotopism, confirming their non-realizability through exhaustive computation.