The number of mates of latin squares of sizes 7 and 8∗
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چکیده
We study the number of mates that a latin square may possess as a function of the size of the square. We performed an exhaustive computer search of all squares of sizes 7 and 8, giving the exact value for the maximum number of mates for squares of these sizes. The squares of size 8 with the maximum number of mates are exactly the Cayley tables of Z2 = Z2×Z2×Z2, and each such square has 70, 272 · 8! mates. We obtain a combinatorial proof that, for every k ≥ 2, the square obtained from a Cayley table of Z2 has a mate. This research was partially supported by NSF grants OCI-1005117 and EPS-0918949.
منابع مشابه
Intersection numbers of Latin squares with their own orthogonal mates
Let J∗(v) be the set of all integers k such that there is a pair of Latin squares L and L′ with their own orthogonal mates on the same v-set, and with L and L′ having k cells in common. In this article we completely determine the set J∗(v) for integers v ≥ 24 and v = 1, 3, 4, 5, 8, 9. For v = 7 and 10 ≤ v ≤ 23, there are only a few cases left undecided for the set J∗(v).
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تاریخ انتشار 2012