A307163 - OEIS

A307163

Minimum number of intercalates in a diagonal Latin square of order n.

0, 0, 0, 12, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0

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OFFSET

1,4

COMMENTS

An intercalate is a 2 X 2 subsquare of a Latin square.

Every diagonal Latin square is a Latin square, so 0 <= a(n) <= A307164(n) <= A092237(n). - Eduard I. Vatutin, Sep 21 2020

Every intercalate is a partial loop and every partial loop is a loop, so 0 <= a(n) <= A307170(n) <= A307166(n). - Eduard I. Vatutin, Oct 19 2020

a(n)=0 for all orders n for which cyclic diagonal Latin squares exist (see A007310) due to all cyclic diagonal Latin squares don't have intercalates. - Eduard I. Vatutin, Aug 07 2023

a(n)=0 for all orders n for which diagonalized cyclic diagonal Latin squares exist (see A372922) due to all diagonalized cyclic diagonal Latin squares don't have intercalates. - Eduard I. Vatutin, Sep 24 2024

a(16) <= 2, a(17) = 0, a(18) <= 9, a(19) = 0, a(20) = 0, a(21) <= 11, a(22) <= 9, a(23) = 0, a(24) <= 2, a(25) = 0, a(26) <= 29, a(27) <= 35, a(28) <= 33. - Eduard I. Vatutin, added Sep 10 2023, updated Oct 05 2025

LINKS

Table of n, a(n) for n=1..15.

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian)

E. I. Vatutin, About the minimum number of intercalates in a diagonal Latin squares of order 9 (in Russian)

E. I. Vatutin, On the inequalities of the minimum and maximum numerical characteristics of diagonal Latin squares for intercalates, loops and partial loops (in Russian)

Eduard I. Vatutin, About the heuristic approximation of the spectrum of number of intercalates in diagonal Latin squares of order 14 (in Russian)

Eduard I. Vatutin, About the minimum number of intercalates in diagonal Latin squares of order 15 (in Russian).

Eduard I. Vatutin, Proving list (best known examples).

E. Vatutin, A. Belyshev, N. Nikitina, and M. Manzuk, Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10, Communications in Computer and Information Science, Vol. 1304, Springer, 2020, pp. 127-146, DOI: 10.1007/978-3-030-66895-2_9.

E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. (in Russian)

Eduard Vatutin, Jia Wei-Ting, Jun Chi Ma, Qiang Miao, Maxim Manzyuk, Natalia Kukushkina, Ilya Kurochkin, and Alexander Albertian, Construction of Intercalate Number Spectra in Latin Squares of Orders 18-28 Using Distributed Parallel Software Implementations, Supercomputing (RuSCDays 2025) 462-474. See references.

Index entries for sequences related to Latin squares and rectangles.