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

1,2

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).

Links

Table of n, a(n) for n=1..13.

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.

Example

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.

Crossrefs

Cf. A309210, A309598, A309599.

Sequence in context: A127932 A006075 A342576 * A241295 A074904 A010304

Adjacent sequences: A370386 A370387 A370388 * A370390 A370391 A370392

Keyword

nonn,more,hard

Author

Eduard I. Vatutin, Feb 17 2024

Status

approved