cp's OEIS Frontend

A diagonal Latin square is a Latin square in which both the main diagonal and main antidiagonal contain each element. - Andrew Howroyd, Oct 05 2020

A Latin square is normalized if in the first row elements come in increasing order. Any diagonal Latin square orthogonal to a given one can be normalized by renaming its elements (which does not break diagonality and orthogonality). - Max Alekseyev, Dec 07 2019

For all orders n>3 there are diagonal Latin squares without orthogonal mates (also known as bachelor squares), so the minimum number of diagonal Latin squares that can be orthogonal to the same diagonal Latin square is zero. For order n=1 the single square is orthogonal to itself. For n=2 and n=3 diagonal Latin squares do not exist (see A274171). For n=6 orthogonal diagonal Latin squares do not exist (see A305571), so a(6)=0. - Eduard I. Vatutin, May 03 2021

a(n) >= A328873(n) - 1. - Eduard I. Vatutin, Mar 29 2021

a(10) >= 10 (Updated). - Eduard I. Vatutin, Apr 27 2018

a(11) >= 32462. - Eduard I. Vatutin from T. Brada, Mar 11 2021

a(12) >= 3855983322. The result belongs to DLS, which has 30192 diagonal transversals. Calculations performed by a volunteer. - Natalia Makarova, Tomáš Brada, Nov 11 2021

a(13) >= 248703. - Natalia Makarova, Tomáš Brada, Apr 29 2021

a(14) >= 307662. - Natalia Makarova, Alex Chernov, Harry White, May 21 2021

a(16) >= 1658880, a(17) >= 2453352, a(18) >= 96, a(19) >= 1383, a(20) >= 995328, a(21) >= 995328, a(22) >= 432000, a(23) >= 525, a(24) >= 345600, a(25) >= 345600, a(26) >= 48, a(27) >= 345600, a(28) >= 663552, a(29) >= 663552, a(30) >= 40320. For values up to a(100), see the specified link "New boundaries for maximum number of normalized orthogonal diagonal Latin squares to one diagonal Latin square". - Natalia Makarova, Alex Chernov, Harry White, Dec 06 2021

Used for getting strong canonical forms (SCFs) of the diagonal Latin squares and for fast enumerating of the diagonal Latin squares based on equivalence classes.

K1 = |C[1]|*f[1] + |C[2]|*f[2] + ... + |C[m]|*f[m],

K2 = K1 * n!,

where m = a(n), number of equivalence classes for X-based filling of diagonals in a diagonal Latin square of order n;

C[i], corresponding equivalence classes with cardinalities |C[i]|, 1 <= i <= m;

f[i], the number of diagonal Latin squares corresponds to the each item from equivalence class C[i], 1 <= i <= m;

K1 = A274171(n), number of diagonal Latin squares of order n with fixed first row;

K2 = A274806(n), number of diagonal Latin squares of order n.

For all t>0 a(2*t) = a(2*t+1). - Eduard I. Vatutin, Aug 21 2020

a(14) >= 5225, a(15) >= 5225. - Natalia Makarova, Sep 12 2020

The number of solutions in an equivalence class with the main diagonal in ascending order is at most 4*2^r*r! where r = floor(n/2). This maximum is achieved for orders n >= 10. - Andrew Howroyd, Mar 27 2023

a(9) <= 48; a(10) <= 1536, a(11) <= 1536, a(12) <= 46080, a(13) <= 7680. - Eduard I. Vatutin, Oct 05 2020, updated Apr 08 2025

A bachelor diagonal Latin square is one with no orthogonal mate.

Used for getting strong canonical forms (SCFs) of the diagonal Latin squares and for fast enumerating of the diagonal Latin squares based on equivalence classes.

For all t > 0, a(2*t) = a(2*t+1).

The diagonal Latin square A has a generalized symmetry (automorphism) if for all cells A[x][y] = v and A[x'][y'] = v' the relation is satisfied: x' = Px[x], y' = Py[y], v' = Pv[v], where Px, Py and Pv — some permutations that are describe generalized symmetry (automorphism). In view of the possibility of an equivalent permutation of the rows and columns of the square and the corresponding transformations in permutations Px, Py and Pv, it is not the types of permutations that are important, but the structure of the multisets L(P) of cycle lengths in them (number of different multisets of order n is A000041). In view of this, codes of various generalized symmetries can be described by the types like (1,1,1) (trivial), (1,16,16) and so on (see example). Diagonal Latin squares with generalized symmetries are rare; usually they have a large number of transversals, orthogonal mates, etc.

a(n) <= A000041(n)^3. - Eduard I. Vatutin, Dec 29 2022

For all orders in which diagonal Latin squares exist a(n) >= 1 due to the existence of the trivial generalized symmetry with code (1,1,1) and Px=Py=Pv=e. - Eduard I. Vatutin, Jan 22 2023

The set of the generalized symmetries is a subset of the generalized symmetries in parastrophic slices, so a(n) <= A358515(n). - Eduard I. Vatutin, Jan 24 2023

Orthogonal diagonal Latin squares are a subset of diagonal Latin squares, so A358394(n) <= a(n). - Eduard I. Vatutin, Jan 25 2023

An orthogonal diagonal Latin square is a square that has at least one orthogonal diagonal mate.

A diagonal Latin square A has a generalized symmetry (automorphism) if for all cells A[x][y] = v and A[x'][y'] = v' the relation is satisfied: x' = Px[x], y' = Py[y], v' = Pv[v], where Px, Py and Pv are some permutations that describe generalized symmetry (automorphism). In view of the possibility of an equivalent permutation of the rows and columns of the square and the corresponding transformations in permutations Px, Py and Pv, it is not the types of permutations that are important, but the structure of the multisets L(P) of cycle lengths in them (number of different multisets of order n is A000041). In view of this, codes of various generalized symmetries can be described by the types like (1,1,1) (trivial), (1,16,16) and so on (see example). Diagonal Latin squares with generalized symmetries are rare; usually they have a large number of transversals, orthogonal mates, etc.

a(8) >= 74, a(9) >= 41, a(10) >= 27.

a(n) <= A000041(n)^3. - Eduard I. Vatutin, Jan 01 2023

For all orders in which orthogonal diagonal Latin squares exist a(n) >= 1 due to the existence of the trivial generalized symmetry with code (1,1,1) and Px=Py=Pv=e. - Eduard I. Vatutin, Jan 22 2023

The set of the generalized symmetries is a subset of the generalized symmetries in parastrophic slices, so a(n) <= A358891(n). - Eduard I. Vatutin, Jan 24 2023

Orthogonal diagonal Latin squares are a subset of diagonal Latin squares, so a(n) <= A357473(n). - Eduard I. Vatutin, Jan 25 2023