Year: 2025
Type: preprint
Abstract: In a Latin square, every row can be interpreted as a permutation, and therefore has a parity (even or odd). We prove that in a uniformly random $n\times n$ Latin square, the $n$ row parities are very well approximated by a sequence of $n$ independent unbiased coin flips: for example, the total variation error of this approximation tends to zero as $n\to\infty$. This resolves a conjecture of Cameron. In fact, we prove a generalisation of Cameron's conjecture for the joint distribution of the row parities, column parities and symbol parities (the latter are defined by the symmetry between rows, columns and symbols of a Latin square). Along the way, we introduce several general techniques for the study of random Latin squares, including a new re-randomisation technique via `stable intercalate switchings', and a new approximation theorem comparing random Latin squares with a certain independent model.
Source: arXiv (Cornell University)
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4417080422
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