Bounds on the number of autotopisms and subsquares of a Latin square |
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Authors: | Joshua Browning Douglas S Stones Ian M Wanless |
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Institution: | 1. School of Mathematical Sciences, Monash University, Monash, VIC, 3800, Australia
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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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