A370389 Number of distinct multisets of cycle lengths in the cell mapping schemes in extended self-orthogonal diagonal Latin squares of order n.

1, 4, 4, 4, 5, 15, 16, 19, 20, 43, 48, 57, 63

Offset: 1

Author

Eduard I. Vatutin, Feb 17 2024

Keywords

• nonn
• more
• hard

Comments

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.

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).

Examples

			For order n=5 there are 5 different multisets of cycle lengths for ESODLS CMS:
1. {1, 1, ..., 1} (25 times) = {1:25};
2. {1:5, 2:10};
3. {1:1, 4:6};
4. {1:1, 2:12};
5. {1:9, 2:8},
so a(5)=5.
		

Links

• Vatutin E.I., About the ESODLS CMS multisets of cycle lengths for orders 11-13 (in Russian).
• Vatutin E.I., Zaikin O.S., Manzuk M.O., and Nikitina N.N., Searching for Orthogonal Latin Squares via Cells Mapping and BOINC-Based Cube-And-Conquer, Communications in Computer and Information Science, 2021, Vol. 1510, pp. 498-512, DOI: 10.1007/978-3-030-92864-3_38.
• Vatutin E.I., Belyshev A.D., Nikitina N.N., and Manzuk M.O., Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares (in Russian), Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16.
• Vatutin E. and Zaikin O., Classification of Cells Mapping Schemes Related to Orthogonal Diagonal Latin Squares of Small Order, Lecture Notes in Computer Science, Vol. 14389, Springer, Cham., 2023, pp. 21-34, DOI: 10.1007/978-3-031-49435-2_2.

Crossrefs

Cf. A309210, A309598, A309599.