A274171 -id:A274171 - cp's OEIS frontend

A358515 Number of types of generalized symmetries in diagonal Latin squares of order n in parastrophic slices.

6, 0, 0, 76, 74, 199, 861

Offset: 1

Eduard I. Vatutin, Nov 20 2022

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). A diagonal Latin square A has a generalized symmetry in parastrophic slices if for all cells A[x][y] = v and A[a'][b'] = c' the relation is satisfied: (a',b',c') = R(Px[x],Py[y],Pv[v]), where R is one of 6 possible parastrophic transformations:
   1. (x,y,v) -> (a,b,c) (trivial).
   2. (x,v,y) -> (a,b,c).
   3. (y,x,v) -> (a,b,c) (transpose).
   4. (y,v,x) -> (a,b,c).
   5. (v,x,y) -> (a,b,c).
   6. (v,y,x) -> (a,b,c).
A set of squares with selected parastrophic transformation R forms one of 6 parastrophic slices. Diagonal Latin squares with a generalized symmetry are a special case of generalized symmetries in parastrophic slice # 1. Diagonal Latin squares with generalized symmetries in parastrophic slices are rare; usually they have a large number of transversals, orthogonal mates, etc.
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) in first parastrophic slice and Px=Py=Pv=e. - Eduard I. Vatutin, Jan 24 2023, updated Mar 25 2023
The set of generalized symmetries is a subset of the generalized symmetries in parastrophic slices, so A357473(n) <= a(n). - Eduard I. Vatutin, Jan 25 2023
Orthogonal diagonal Latin squares are a subset of diagonal Latin squares, so A358891(n) <= a(n). - Eduard I. Vatutin, Jan 28 2023

			For order n=4 there are 5 different multisets L(P) with codes listed below in format "code - multiset":
  1 - {1,1,1,1},
  2 - {1,1,2},
  3 - {1,3},
  4 - {2,2},
  5 - {4}.
Diagonal Latin squares of order n=4 have a(4)=76 different types of generalized symmetries in parastrophic slices.
Slice 1 (10 generalized symmetries), R=(x,y,v):
  1. A=0123321010322301 (string representation of the square), Px=[0,1,2,3], Py=[0,1,2,3], Pv=[0,1,2,3] (trivial generalized symmetry), L(Px)={1,1,1,1,1}, L(Py)={1,1,1,1,1}, L(Pv)={1,1,1,1,1}, generalized symmetry type 1-(1,1,1).
  2. A=0123321010322301, Px=[0,1,2,3], Py=[1,0,3,2], Pv=[1,0,3,2], L(Px)={1,1,1,1}, L(Py)={2,2}, L(Pv)={2,2}, generalized symmetry type 1-(1,4,4).
  ...
  10. A=0123321010322301, Px=[1,2,3,0], Py=[2,3,1,0], Pv=[1,0,2,3], L(Px)={4}, L(Py)={4}, L(Pv)={1,1,2}, generalized symmetry type 1-(5,5,2).
Slice 2 (10 generalized symmetries), R=(x,v,y):
  11. A=0123321010322301, Px=[0,1,2,3], Py=[0,1,2,3], Pv=[0,1,2,3], L(Px)={1,1,1,1,1}, L(Py)={1,1,1,1,1}, L(Pv)={1,1,1,1,1}, generalized symmetry type 2-(1,1,1).
  12. A=0123321010322301, Px=[0,1,2,3], Py=[1,0,3,2], Pv=[1,0,3,2], L(Px)={1,1,1,1}, L(Py)={2,2}, L(Pv)={2,2}, generalized symmetry type 2-(1,4,4).
  ...
  20. A=0123321010322301, Px=[1,2,3,0], Py=[2,3,1,0], Pv=[1,0,2,3], L(Px)={4}, L(Py)={4}, L(Pv)={1,1,2}, generalized symmetry type 2-(5,5,2).
Slice 3 (14 generalized symmetries).
Slice 4 (14 generalized symmetries).
Slice 5 (14 generalized symmetries).
Slice 6 (14 generalized symmetries).
Total 10+10+14+14+14+14=76 generalized symmetries in parastrophic slices.

Eduard I. Vatutin, About the number of types of generalized symmetries in diagonal Latin squares of orders 1-7.
Eduard I. Vatutin, Proving lists (4, 5, 6, 7), Jul 28 2022.
Index entries for sequences related to Latin squares and rectangles.

Cf. A000041, A274171, A287649, A287650, A293777, A357473, A358394, A358891.

a(n) <= 6*A000041(n)^3. - Eduard I. Vatutin, Dec 29 2022

A358891 Number of types of generalized symmetries in orthogonal diagonal Latin squares of order n in parastrophic slices.

Original entry on oeis.org

6, 0, 0, 76, 44, 0, 145

Offset: 1

Eduard I. Vatutin, Dec 05 2022

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). A diagonal Latin square A has a generalized symmetry in parastrophic slices if for all cells A[x][y] = v and A[a'][b'] = c' the relation is satisfied: (a',b',c') = R(Px[x],Py[y],Pv[v]), where R is one of 6 possible parastrophic transformations:
   1. (x,y,v) -> (a,b,c) (trivial).
   2. (x,v,y) -> (a,b,c).
   3. (y,x,v) -> (a,b,c) (transpose).
   4. (y,v,x) -> (a,b,c).
   5. (v,x,y) -> (a,b,c).
   6. (v,y,x) -> (a,b,c).
A set of squares with selected parastrophic transformation R forms one of 6 parastrophic slices. Diagonal Latin squares with a generalized symmetry are a special case of generalized symmetries in parastrophic slice # 1. Diagonal Latin squares with generalized symmetries in parastrophic slices are rare; usually they have a large number of transversals, orthogonal mates, etc.
a(8) >= 3874, a(9) >= 8907, a(10) >= 3592.
a(n) <= 6*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) in first parastrophic slice and Px=Py=Pv=e. - Eduard I. Vatutin, Jan 24 2023, updated Mar 25 2023
The set of generalized symmetries is a subset of the generalized symmetries in parastrophic slices, so A358394(n) <= a(n). - Eduard I. Vatutin, Jan 25 2023
Orthogonal diagonal Latin squares are a subset of diagonal Latin squares, so a(n) <= A358515(n). - Eduard I. Vatutin, Jan 28 2023

			For order n=4 there are 5 different multisets L(P) with codes listed below in format "code - multiset":
  1 - {1,1,1,1},
  2 - {1,1,2},
  3 - {1,3},
  4 - {2,2},
  5 - {4}.
Diagonal Latin squares of order n=4 have a(4)=76 different types of generalized symmetries in parastrophic slices.
Slice 1 (10 generalized symmetries), R=(x,y,v):
  1. A=0123321010322301 (string representation of the square), Px=[0,1,2,3], Py=[0,1,2,3], Pv=[0,1,2,3] (trivial generalized symmetry), L(Px)={1,1,1,1,1}, L(Py)={1,1,1,1,1}, L(Pv)={1,1,1,1,1}, generalized symmetry type 1-(1,1,1).
  2. A=0123321010322301, Px=[0,1,2,3], Py=[1,0,3,2], Pv=[1,0,3,2], L(Px)={1,1,1,1}, L(Py)={2,2}, L(Pv)={2,2}, generalized symmetry type 1-(1,4,4).
  ...
  10. A=0123321010322301, Px=[1,2,3,0], Py=[2,3,1,0], Pv=[1,0,2,3], L(Px)={4}, L(Py)={4}, L(Pv)={1,1,2}, generalized symmetry type 1-(5,5,2).
Slice 2 (10 generalized symmetries), R=(x,v,y):
  11. A=0123321010322301, Px=[0,1,2,3], Py=[0,1,2,3], Pv=[0,1,2,3], L(Px)={1,1,1,1,1}, L(Py)={1,1,1,1,1}, L(Pv)={1,1,1,1,1}, generalized symmetry type 2-(1,1,1).
  12. A=0123321010322301, Px=[0,1,2,3], Py=[1,0,3,2], Pv=[1,0,3,2], L(Px)={1,1,1,1}, L(Py)={2,2}, L(Pv)={2,2}, generalized symmetry type 2-(1,4,4).
  ...
  20. A=0123321010322301, Px=[1,2,3,0], Py=[2,3,1,0], Pv=[1,0,2,3], L(Px)={4}, L(Py)={4}, L(Pv)={1,1,2}, generalized symmetry type 2-(5,5,2).
Slice 3 (14 generalized symmetries).
Slice 4 (14 generalized symmetries).
Slice 5 (14 generalized symmetries).
Slice 6 (14 generalized symmetries).
Total 10+10+14+14+14+14=76 generalized symmetries in parastrophic slices.

Eduard I. Vatutin, About the number of types of generalized symmetries in diagonal Latin squares of orders 1-7.
Eduard I. Vatutin, Proving lists (4, 5, 7, 8, 9, 10), Jul 31 2022
Index entries for sequences related to Latin squares and rectangles.

Cf. A000041, A274171, A287649, A287650, A293777, A357473, A358515, A358394.

A337303 Number of X-based filling of diagonals in a diagonal Latin square of order n.

Original entry on oeis.org

1, 1, 0, 0, 96, 480, 57600, 403200, 191600640, 1724405760, 1597368729600, 17571056025600, 28378507272192000, 368920594538496000, 952903592436341145600, 14293553886545117184000, 55442575636536644075520000, 942523785821122949283840000, 5231730206388249282710863872000

Offset: 0

Eduard I. Vatutin, Aug 22 2020

nonn

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.

			One of the 96 X-based fillings of diagonals of a diagonal Latin square for order n=4:
1 . . 0
. 0 1 .
. 3 2 .
2 . . 3

Andrew Howroyd, Table of n, a(n) for n = 0..100
S. Kochemazov, O. Zaikin, E. Vatutin E., and A. Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, Journal of Integer Sequences. Vol. 23. Iss. 1. 2020. Article 20.1.2.
E. I. Vatutin, About the number of X-based fillings of diagonals in a diagonal Latin squares of orders 1-15 (in Russian).
E. I. Vatutin, A. D. Belyshev, N. N. Nikitina, and M. O. Manzuk, Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares, Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16 (in Russian).
Index entries for sequences related to Latin squares and rectangles.

Cf. A000316, A000459, A309283, A274171, A337302.

\\ here b(n) is A000459.
b(n) = {sum(m=0, n, sum(k=0, n-m, (-1)^k * binomial(n, k) * binomial(n-k, m) * 2^(2*k+m-n) * (2*n-2*m-k)! )); }
a(n) = {2^(n\2) * b(n\2) * n!} \\ Andrew Howroyd, Mar 26 2023

a(n) = A337302(n)*n!.

a(n) = n!*A000316(floor(n/2)). - Andrew Howroyd, Mar 26 2023

a(0)=1 prepended and terms a(16) and beyond from Andrew Howroyd, Mar 26 2023

cp's OEIS Frontend

A358515 Number of types of generalized symmetries in diagonal Latin squares of order n in parastrophic slices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A358891 Number of types of generalized symmetries in orthogonal diagonal Latin squares of order n in parastrophic slices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A337303 Number of X-based filling of diagonals in a diagonal Latin square of order n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions