On the probability that a random Latin square is all-even
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× n Latin square, the probability that all rows are even is of the form 2−n+o(n). This fills a gap left by recent work of Kwan, Petrova and Sawhney: they proved that the row parities behave like a sequence of independent coin flips in a variety of different senses, but they were not able to lower-bound the all-even probability, as their approach had a small number of "exceptional rows" whose parities they could not control. In fact, we prove a more general theorem on the probability of any possible prescription of row parities, column parities and symbol parities. Our approach extends the ideas of Kwan, Petrova and Sawhney, introducing a new way to "bootstrap" intercalate switchings to control more complex switching operations.
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BibTeX
@misc{greenhill2026probability,
title = {{On the probability that a random Latin square is all-even}},
author = {Catherine Greenhill and Lenka Kopfová and Matthew Kwan and Kalina Petrova},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13487},
eprint = {2610.13487},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13487},
}
RIS
TY - UNPB
TI - On the probability that a random Latin square is all-even
AU - Catherine Greenhill
AU - Lenka Kopfová
AU - Matthew Kwan
AU - Kalina Petrova
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13487
UR - https://arxiv.org/abs/2610.13487
AB - 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× n Latin square, the probability that all rows are even is of the form 2^−n+o(n). This fills a gap left by recent work of Kwan, Petrova and Sawhney: they proved that the row parities behave like a sequence of independent coin flips in a variety of different senses, but they were not able to lower-bound the all-even probability, as their approach had a small number of "exceptional rows" whose parities they could not control. In fact, we prove a more general theorem on the probability of any possible prescription of row parities, column parities and symbol parities. Our approach extends the ideas of Kwan, Petrova and Sawhney, introducing a new way to "bootstrap" intercalate switchings to control more complex switching operations.
ER -