Bounds on the number of autotopisms and subsquares of a Latin square

Abstract

A subsquare of a Latin square L is a submatrix that is also a Latin square. An autotopism of L is a triplet of permutations (α, β, γ) such that L is unchanged after the rows are permuted by α, the columns are permuted by β and the symbols are permuted by γ. Let n!(n−1)!R n be the number of n×n Latin squares. We show that an n×n Latin square has at most n O(log k) subsquares of order k and admits at most n O(log n) autotopisms. This enables us to show that {ie11-1} divides R n for all primes p. We also extend a theorem by McKay and Wanless that gave a factorial divisor of R n , and give a new proof that R p ≠1 (mod p) for prime p.

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Correspondence to Joshua Browning.

Additional information

Supported by ARC grants DP0662946 and DP1093320. Stones also partially supported by NSFC grant 61170301.

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Browning, J., Stones, D.S. & Wanless, I.M. Bounds on the number of autotopisms and subsquares of a Latin square. Combinatorica 33, 11–22 (2013). https://doi.org/10.1007/s00493-013-2809-1

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Mathematics Subject Classification (2000)

  • 05B15
  • 20D45
  • 20N05