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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A357473 Number of types of generalized symmetries in diagonal Latin squares of order n.

Original entry on oeis.org

1, 0, 0, 10, 8, 12, 12
Offset: 1

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Author

Eduard I. Vatutin, Sep 29 2022

Keywords

Comments

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

Examples

			For order n=5 there are 7 different multisets L(P) with codes listed below in format "code - multiset":
  1 - {1,1,1,1,1},
  2 - {1,1,1,2},
  3 - {1,1,3},
  4 - {1,2,2},
  5 - {1,4},
  6 - {2,3},
  7 - {5}.
Diagonal Latin squares of order n=5 has a(5)=8 different types of generalized symmetries:
1. A=0123442301341201304220413 (string representation of the square), Px=[0,1,2,3,4], Py=[0,1,2,3,4], Pv=[0,1,2,3,4] (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).
2. A=0123442301341201304220413, Px=[0,1,2,3,4], Py=[1,3,0,4,2], Pv=[1,3,0,4,2], L(Px)={1,1,1,1,1}, L(Py)={5}, L(Pv)={5}, generalized symmetry type (1,7,7).
3. A=0123442013143203014223401, Px=[0,3,2,4,1], Py=[1,4,2,3,0], Pv=[1,4,2,3,0], L(Px)={1,1,3}, L(Py)={1,1,3}, L(Pv)={1,1,3}, generalized symmetry type (3,3,3).
4. A=0123442301341201304220413, Px=[0,2,1,4,3], Py=[0,2,1,4,3], Pv=[0,2,1,4,3], L(Px)={1,2,2}, L(Py)={1,2,2}, L(Pv)={1,2,2}, generalized symmetry type (4,4,4).
5. A=0123442301341201304220413, Px=[0,3,4,2,1], Py=[0,3,4,2,1], Pv=[0,3,4,2,1], L(Px)={1,4}, L(Py)={1,4}, L(Pv)={1,4}, generalized symmetry type (5,5,5).
6. A=0123442301341201304220413, Px=[1,3,0,4,2], Py=[0,1,2,3,4], Pv=[4,2,3,0,1], L(Px)={5}, L(Py)={1,1,1,1,1}, L(Pv)={5}, generalized symmetry type (7,1,7).
7. A=0123442301341201304220413, Px=[1,3,0,4,2], Py=[3,4,1,2,0], Pv=[0,1,2,3,4], L(Px)={5}, L(Py)={5}, L(Pv)={1,1,1,1,1}, generalized symmetry type (7,7,1).
8. A=0123442301341201304220413, Px=[1,3,0,4,2], Py=[1,3,0,4,2], Pv=[2,0,4,1,3], L(Px)={5}, L(Py)={5}, L(Pv)={5}, generalized symmetry type (7,7,7).

Links

Crossrefs

Cf. A000041, A274171, A287649, A287650, A293777, A358515, A358394, A358891.

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

Original entry on oeis.org

6, 0, 0, 76, 74, 199, 861
Offset: 1

Views

Author

Eduard I. Vatutin, Nov 20 2022

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Formula

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

Views

Author

Eduard I. Vatutin, Dec 05 2022

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A000041, A274171, A287649, A287650, A293777, A357473, A358515, A358394.
Showing 1-3 of 3 results.

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