A342998 -id:A342998 - cp's OEIS frontend

A287647 Minimum number of diagonal transversals in a diagonal Latin square of order n.

1, 0, 0, 4, 1, 2, 0, 0, 0

Offset: 1

Eduard I. Vatutin, May 29 2017

A007016 is an upper bound for the number of diagonal transversals in a Latin square: a(n) <= A287648(n) <= A007016(n). - Eduard I. Vatutin, Jan 02 2020
From Eduard I. Vatutin, Apr 26 2021: (Start)
A diagonal Latin square is a Latin square in which both the main diagonal and main antidiagonal contain each element.
A diagonal transversal is a transversal that includes exactly one element from the main diagonal and exactly one from the antidiagonal. For squares of odd orders, these elements can coincide at the intersection of the diagonals.
All cyclic diagonal Latin squares (see A338562) are diagonal Latin squares, so a(n) <= A342998((n-1)/2). (End)
a(10) <= 3, a(11) <= 43, a(12) = 0, a(13) <= 4756, a(14) <= 1446, a(15) <= 15510, a(16) <= 898988, a(17) <= 12058840, a(18) <= 82577875, a(19) <= 592174879, a(20) <= 4488686380. - Eduard I. Vatutin, Sep 26 2021, updated Jan 20 2025

			From _Eduard I. Vatutin_, Apr 26 2021: (Start)
For example, the diagonal Latin square
  0 1 2 3
  3 2 1 0
  1 0 3 2
  2 3 0 1
has 4 diagonal transversals:
  0 . . .    . 1 . .    . . 2 .    . . . 3
  . . 1 .    . . . 0    3 . . .    . 2 . .
  . . . 2    . . 3 .    . 0 . .    1 . . .
  . 3 . .    2 . . .    . . . 1    . . 0 .
In addition there are 4 other transversals that are not diagonal transversals and are therefore not included here. (End)

E. I. Vatutin, Discussion about properties of diagonal Latin squares, forum.boinc.ru (in Russian)
E. I. Vatutin, About the minimal number of diagonal transversals in a diagonal Latin squares of order 9 (in Russian)
Eduard I. Vatutin, Best known examples.
E. I. Vatutin, S. E. Kochemazov and O. S. Zaikin, Estimating of combinatorial characteristics for diagonal Latin squares, Recognition — 2017 (2017), pp. 98-100 (in Russian)
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin and S. Yu. Valyaev, Enumerating the Transversals for Diagonal Latin Squares of Small Order. CEUR Workshop Proceedings. Proceedings of the Third International Conference BOINC-based High Performance Computing: Fundamental Research and Development (BOINC:FAST 2017). Vol. 1973. Technical University of Aachen, Germany, 2017. pp. 6-14. urn:nbn:de:0074-1973-0.
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin and S. Yu. Valyaev, Using Volunteer Computing to Study Some Features of Diagonal Latin Squares. Open Engineering. Vol. 7. Iss. 1. 2017. pp. 453-460. DOI: 10.1515/eng-2017-0052
E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, S. Yu. Valyaev and V. S. Titov, Estimating the Number of Transversals for Diagonal Latin Squares of Small Order, Telecommunications. 2018. No. 1. pp. 12-21 (in Russian).
Eduard I. Vatutin, Natalia N. Nikitina and Maxim O. Manzuk, First results of an experiment on studying the properties of DLS of order 9 in the volunteer distributed computing projects Gerasim@Home and RakeSearch (in Russian).
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)
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, A. M. Albertyan, A. V. Kripachev, and A. I. Pykhtin, Methods for getting spectra of fast computable numerical characteristics of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 19-23. (in Russian)
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, Estimation of the Cardinalities of the Spectra of Fast-computable Numerical Characteristics for Diagonal Latin Squares of Orders N>9 (in Russian) // Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2022. pp. 314-315.
Index entries for sequences related to Latin squares and rectangles.

Cf. A287644, A287645, A287648, A342998, A345370.

a(8) added by Eduard I. Vatutin, Oct 29 2017

a(9) added by Eduard I. Vatutin, Sep 20 2020

A342997 Maximum number of diagonal transversals in a cyclic diagonal Latin square of order 2n+1.

Original entry on oeis.org

1, 0, 5, 27, 0, 4665, 131106, 0, 204995269, 11254190082

Offset: 0

Eduard I. Vatutin, Apr 02 2021

A cyclic Latin square is a Latin square in which row i is obtained by cyclically shifting row i-1 by d places (see A338562, A123565 and A341585).
Cyclic diagonal Latin squares do not exist for even n.
All cyclic diagonal Latin squares are diagonal Latin squares, so a((n-1)/2) <= A287648(n).
All diagonal transversals are transversals, so a(n) <= A006717(n).
A342998 <= a(n).

			For n=2 one of the best cyclic diagonal Latin squares of order 5
  0 1 2 3 4
  2 3 4 0 1
  4 0 1 2 3
  1 2 3 4 0
  3 4 0 1 2
has a(2)=5 diagonal transversals:
  0 . . . .   . 1 . . .   . . 2 . .   . . . 3 .   . . . . 4
  . . 4 . .   . . . 0 .   . . . . 1   2 . . . .   . 3 . . .
  . . . . 3   4 . . . .   . 0 . . .   . . 1 . .   . . . 2 .
  . 2 . . .   . . 3 . .   . . . 4 .   . . . . 0   1 . . . .
  . . . 1 .   . . . . 2   3 . . . .   . 4 . . .   . . 0 . .

Eduard I. Vatutin, Enumerating the diagonal transversals for cyclic diagonal Latin squares of orders 1-19 (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A006717, A123565, A287648, A338562, A341585, A342998.

A348212 Number of transversals in a cyclic diagonal Latin square of order 2n+1.

Original entry on oeis.org

1, 0, 15, 133, 0, 37851, 1030367, 0, 1606008513, 87656896891, 0, 452794797220965, 41609568918940625

Offset: 1

Eduard I. Vatutin, Oct 07 2021

All cyclic diagonal Latin squares of order n have same number of transversals. A similar statement for diagonal transversals is not true (see A342998 and A342997).
All broken diagonals and antidiagonals of cyclic Latin squares are transversals, so a(n) >= 2*n for all n > 1 for which cyclic diagonal Latin squares exist. - Eduard I. Vatutin, Mar 22 2022
All cyclic diagonal Latin squares are diagonal Latin squares, so A287645(2n+1) <= a(n) <= A287644(2n+1) for all orders in which cyclic diagonal Latin squares exist. - Eduard I. Vatutin, Mar 23 2022

			A cyclic diagonal Latin square of order 5
  0 1 2 3 4
  2 3 4 0 1
  4 0 1 2 3
  1 2 3 4 0
  3 4 0 1 2
has a(3)=15 transversals:
  0 . . . .   0 . . . .   . 1 . . .         . . . . 4
  . 3 . . .   . . . . 1   2 . . . .         . 3 . . .
  . . 1 . .   . . . 2 .   . . . . 3         . . . 2 .
  . . . 4 .   . . 3 . .   . . . 4 .         1 . . . .
  . . . . 2   . 4 . . .   . . 0 . .   ...   . . 0 . .

Eduard I. Vatutin, About the number of transversals in cyclic Latin and cyclic diagonal Latin squares (in Russian).
Index entries for sequences related to Latin squares and rectangles.

Cf. A006717, A011655, A287644, A287645, A338562, A342997, A342998.

a(n) = A006717(n) * A011655(n+1).

A366332 Minimum number of diagonal transversals in a semicyclic diagonal Latin square of order 2n+1.

Original entry on oeis.org

1, 0, 5, 27, 0, 4523, 127339, 0, 204330233, 11232045257, 0

Offset: 0

Eduard I. Vatutin, Oct 07 2023

A horizontally semicyclic diagonal Latin square is a square where each row r(i) is a cyclic shift of the first row r(0) by some value d(i) (see example). Similarly, a vertically semicyclic diagonal Latin square is a square where each column c(i) is a cyclic shift of the first column c(0) by some value d(i).

			Example of horizontally semicyclic diagonal Latin square of order 13:
   0  1  2  3  4  5  6  7  8  9 10 11 12
   2  3  4  5  6  7  8  9 10 11 12  0  1  (d=2)
   4  5  6  7  8  9 10 11 12  0  1  2  3  (d=4)
   9 10 11 12  0  1  2  3  4  5  6  7  8  (d=9)
   7  8  9 10 11 12  0  1  2  3  4  5  6  (d=7)
  12  0  1  2  3  4  5  6  7  8  9 10 11  (d=12)
   3  4  5  6  7  8  9 10 11 12  0  1  2  (d=3)
  11 12  0  1  2  3  4  5  6  7  8  9 10  (d=11)
   6  7  8  9 10 11 12  0  1  2  3  4  5  (d=6)
   1  2  3  4  5  6  7  8  9 10 11 12  0  (d=1)
   5  6  7  8  9 10 11 12  0  1  2  3  4  (d=5)
  10 11 12  0  1  2  3  4  5  6  7  8  9  (d=10)
   8  9 10 11 12  0  1  2  3  4  5  6  7  (d=8)

Eduard I. Vatutin, About the spectra of numerical characteristics of different types of cyclic diagonal Latin squsres (in Russian).
Eduard I. Vatutin, About the spectra of numerical characteristics of semicyclic diagonal Latin squares of order 17 (in Russian).
Eduard I. Vatutin, About the spectra of numerical characteristics of semicyclic diagonal Latin squares of order 19 (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A071607, A342990, A342997, A342998, A348212, A366331.

cp's OEIS Frontend

A287647 Minimum number of diagonal transversals in a diagonal Latin square of order n.

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A342997 Maximum number of diagonal transversals in a cyclic diagonal Latin square of order 2n+1.

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A348212 Number of transversals in a cyclic diagonal Latin square of order 2n+1.

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A366332 Minimum number of diagonal transversals in a semicyclic diagonal Latin square of order 2n+1.

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A376587 Minimum number of diagonal transversals in diagonalized cyclic diagonal Latin squares of order 2n+1.

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