## Abstract

Let *L*(*n*) be the number of Latin squares of order *n*, and let *L*
^{even}(*n*) and *L*
^{odd}(*n*) be the number of even and odd such squares, so that *L*(*n*)=*L*
^{even}(*n*)+*L*
^{odd}(*n*). The Alon-Tarsi conjecture states that *L*
^{even}(*n*) ≠ *L*
^{odd}(*n*) when *n* is even (when *n* is odd the two are equal for very simple reasons). In this short note we prove that

, thus establishing the conjecture that the number of even and odd Latin squares, while conjecturally not equal in even dimensions, are equal to leading order asymptotically. Two proofs are given: both proceed by applying a differential operator to an exponential integral over SU(*n*). The method is inspired by a recent result of Kumar-Landsberg.

This is a preview of subscription content, access via your institution.

## References

- [1]
N. Alon and M. Tarsi: Colorings and orientations of graphs,

*Combinatorica***12**(1992), 125–134. - [2]
M. Creutz: On invariant integration over SU(N),

*J. Math. Phys.***19**(1978), 2043–2046. - [3]
A. A. Drisko: On the number of even and odd Latin squares of order p+1,

*Adv. Math.***128**(1997), 20–35. - [4]
D. G. Glynn: The conjectures of Alon-Tarsi and Rota in dimension prime minus one,

*SIAM J. Discrete Math.***24**(2010), 394–399. - [5]
S. Kumar and J. M. Landsberg: Connections between conjectures of Alon-Tarsi, Hadamard-Howe, and integrals over the special unitary group, See http://arxiv.org/abs/1410.8585, preprint (2014).

- [6]
D. S. Stones and I. M. Wanless: How not to prove the Alon-Tarsi conjecture,

*Nagoya Math. J.***205**(2012), 1–24. - [7]
J. H. van Lint and R. M. Wilson:

*A course in combinatorics*, Cambridge University Press, Cambridge, second edition, 2001.

## Author information

### Affiliations

### Corresponding author

## Rights and permissions

## About this article

### Cite this article

Alpoge, L. Square-root cancellation for the signs of Latin squares.
*Combinatorica* **37, **137–142 (2017). https://doi.org/10.1007/s00493-015-3373-7

Received:

Published:

Issue Date:

### Mathematics Subject Classification (2000)

- 05B15
- 05A16