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%I A370389 #11 Mar 12 2024 15:34:52
%S A370389 1,4,4,4,5,15,16,19,20,43,48,57,63
%N A370389 Number of distinct multisets of cycle lengths in the cell mapping schemes in extended self-orthogonal diagonal Latin squares of order n.
%C A370389 A cells mapping scheme (CMS) for an ordered pair (A,B) of Latin squares is a permutation p of N^2 integer numbers from 0 to N^2-1 such that p[i] = j, 0 <= i, j <= N^2-1 iff A[i] = B[j] (square’s elements are listed left-to-right and top-to-bottom in the string representation). Used for getting ESODLS (see A309210). Structure of the multiset of cycle lengths in the CMS provides cycle of ESODLS with length equal to the least common multiple of cycle lengths in the CMS.
%C A370389 An extended self-orthogonal diagonal Latin square (ESODLS) is a diagonal Latin square that has an orthogonal diagonal Latin square from the same main class (see A309598).
%H A370389 Vatutin E.I., <a href="https://vk.com/wall162891802_2661">About the ESODLS CMS multisets of cycle lengths for orders 11-13</a> (in Russian).
%H A370389 Vatutin E.I., Zaikin O.S., Manzuk M.O., and Nikitina N.N., <a href="https://doi.org/10.1007/978-3-030-92864-3_38">Searching for Orthogonal Latin Squares via Cells Mapping and BOINC-Based Cube-And-Conquer</a>, Communications in Computer and Information Science, 2021, Vol. 1510, pp. 498-512, DOI: 10.1007/978-3-030-92864-3_38.
%H A370389 Vatutin E.I., Belyshev A.D., Nikitina N.N., and Manzuk M.O., <a href="https://doi.org/10.31044/1684-2588-2023-0-1-2-16">Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares</a> (in Russian), Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16.
%H A370389 Vatutin E. and Zaikin O., <a href="https://doi.org/10.1007/978-3-031-49435-2_2">Classification of Cells Mapping Schemes Related to Orthogonal Diagonal Latin Squares of Small Order</a>, Lecture Notes in Computer Science, Vol. 14389, Springer, Cham., 2023, pp. 21-34, DOI: 10.1007/978-3-031-49435-2_2.
%e A370389 For order n=5 there are 5 different multisets of cycle lengths for ESODLS CMS:
%e A370389 1. {1, 1, ..., 1} (25 times) = {1:25};
%e A370389 2. {1:5, 2:10};
%e A370389 3. {1:1, 4:6};
%e A370389 4. {1:1, 2:12};
%e A370389 5. {1:9, 2:8},
%e A370389 so a(5)=5.
%Y A370389 Cf. A309210, A309598, A309599.
%K A370389 nonn,more,hard
%O A370389 1,2
%A A370389 _Eduard I. Vatutin_, Feb 17 2024

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