Transversals in quasirandom latin squares

TL;DR

This paper refines the asymptotic count of transversals in quasirandom latin squares, extending results to a broader class including random squares and quasirandom groups, using a variant of the circle method.

Contribution

It provides a sharper asymptotic estimate for the number of transversals in quasirandom latin squares, applicable to a wide class of such squares.

Findings

Almost all quasirandom latin squares have approximately (e^{-1/2}) n!^2 / n^n transversals.

The method applies to random latin squares and multiplication tables of quasirandom groups.

The circle method variant effectively counts transversals under quasirandomness conditions.

Abstract

A transversal in an $n \times n$ latin square is a collection of $n$ entries not repeating any row, column, or symbol. Kwan showed that almost every $n \times n$ latin square has $\bigl((1 + o(1)) n / e^2\bigr)^n$ transversals as $n \to \infty$. Using a loose variant of the circle method we sharpen this to $(e^{-1/2} + o(1)) n!^2 / n^n$. Our method works for all latin squares satisfying a certain quasirandomness condition, which includes both random latin squares with high probability as well as multiplication tables of quasirandom groups.