A132206 -id:A132206 - cp's OEIS frontend

A100540 Total number of Latin 4-dimensional hypercubes (Latin polyhedra) of order n.

Original entry on oeis.org

1, 2, 48, 36972288, 52260618977280

Offset: 1

Toru Ito (t_ito(AT)mue.biglobe.ne.jp), Nov 28 2004

T. Ito, Method, equipment,program and storage media for producing tables, Publication number JP2004-272104A, Japan Patent Office (in Japanese).

B. D. McKay and I. M. Wanless, A census of small latin hypercubes, SIAM J. Discrete Math. 22, (2008) 719-736.

Cf. A100539, A132205, A132206.

A row of the array in A249026.

a(5) from Ian Wanless, May 01 2008

Edited by N. J. A. Sloane, Dec 05 2009 at the suggestion of Vladeta Jovovic

A249026 Array read by antidiagonals upwards: T(d,n) = number of d-dimensional permutations of n letters (d >= 0, n >= 1).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 6, 4, 1, 2, 12, 24, 5, 1, 2, 24, 576, 120, 6, 1, 2, 48, 55296, 161280, 720, 7, 1, 2, 96, 36972288, 2781803520, 812851200, 5040, 8, 1, 2, 192, 6268637952000, 52260618977280, 994393803303936000, 61479419904000, 40320, 9

Offset: 0

N. J. A. Sloane, Oct 23 2014

By definition, this is the number of  nXnXnX...Xn = n^(d+1) arrays of 0's and 1's with exactly one 1 in each row, column, ..., line, ... .
An ordinary permutation is the case d = 1 (ordinary matrices with a single 1 in each row and column).
Rows d=2,3,... correspond to Latin squares, cubes, etc.

			The array begins:
d\n: 1, 2, 3,  4,  5,   6,   7,    8,     9,      10,      11,
--------------------------------------------------------------
0:   1, 2, 3,  4,  5,   6,   7,    8,     9,      10,      11,
1:   1, 2, 6,  24, 120, 720, 5040, 40320, 362880, 3628800, 39916800,  ...
2:   1, 2, 12, 576, 161280, 812851200, 61479419904000, 108776032459082956800,...
3:   1, 2, 24, 55296, 2781803520, 994393803303936000, ...
4:   1, 2, 48, 36972288, 52260618977280, ...
5:   1, 2, 96, 6268637952000, 2010196727432478720, ...
6:   1, 2, 192, ...
7:   1, 2, 384, ...
8:   1, 2, 768, ...
...

Linial, Nathan, and Zur Luria, An upper bound on the number of high-dimensional permutations, arXiv preprint arXiv:1106.0649 [math.CO], (2011).
Linial, Nathan, and Zur Luria, An upper bound on the number of high-dimensional permutations, Combinatorica, 34 (2014), 471-486.

Rows: A000142, A002860, A098679, A100540, A132206.
Column 4 = A249028.
See A249027 for another version.

A132205 Number of reduced Latin 5-dimensional hypercubes (Latin polyhedra) of order n.

Original entry on oeis.org

1, 1, 1, 201538000, 50490811256

Offset: 1

Toru Ito (to-itou(AT)ipa.go.jp), Nov 06 2007

Latin 5-dimensional hypercubes (Latin polyhedra) are a generalization of Latin cube and Latin square. a(4) computed on Dec 01 2002.

T. Ito, Method, equipment, program and storage media for producing tables, Publication number JP2004-272104A, Japan Patent Office (written in Japanese).
Kenji Ohkuma, Atsuhiro Yamagishi and Toru Ito, Cryptography Research Group Technical report, IT Security Center, Information-Technology Promotion Agency, JAPAN.

B. D. McKay and I. M. Wanless, A census of small latin hypercubes, SIAM J. Discrete Math. 22, (2008) 719-736.

Cf. A100540, A132206.

a(5) from Ian Wanless, May 01 2008

Edited by N. J. A. Sloane, Dec 05 2009 at the suggestion of Vladeta Jovovic

A211215 Total number of Latin n-dimensional hypercubes of order 4; labeled n-ary quasigroups of order 4.

Original entry on oeis.org

4, 24, 576, 55296, 36972288, 6268637952000, 80686060158523011084288, 4465185218736554544676917926460256725000192, 4558271384916189349044295395852008182480786230841798008741684281906576963885826048

Offset: 0

Denis S. Krotov and Vladimir N. Potapov, Apr 06 2012

The values are calculated recursively, based on the characterization by 2009. The number a(5) was found before (2001 and, independently, later works) by exhaustive computer-aided classification of the objects.

T. Ito, Creation Method of Table, Creation Apparatus, Creation Program and Program Storage Medium, U.S. Patent application 20040243621, Dec 02 2004.

Denis S. Krotov and Vladimir N. Potapov, On the reconstruction of N-quasigroups of order 4 and the upper bounds on their numbers, Proc. Conference devoted to the 90th anniversary of Alexei A. Lyapunov (Novosibirsk, Russia, October 8-11, 2001), 2001.
Denis S. Krotov and Vladimir N. Potapov, n-Ary Quasigroups of Order 4, arXiv:math/0701519 [math.CO], 2007-2008; SIAM J. Discrete Math. 23:2 (2009), 561-570.
B. D. McKay and I. M. Wanless, A census of small latin hypercubes, SIAM J. Discrete Math. 22, (2008) 719-736.
Vladimir N. Potapov and Denis S. Krotov, On the number of n-ary quasigroups of finite order, arXiv:0912.5453 [math.CO], 2009-2016; Discrete Mathematics and Applications, 21:5-6 (2011), 575-586.

Cf. A002860, A098679, A100540, A132206.

Python
```
# See A211214.
```

a(n) = 4*6^n * A211214(n).

A249027 Array read by antidiagonals upwards: T(d,n) = number of d-dimensional permutations of n letters (d >= 1, n >= 1).

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 12, 24, 1, 2, 24, 576, 120, 1, 2, 48, 55296, 161280, 720, 1, 2, 96, 36972288, 2781803520, 812851200, 5040, 1, 2, 192, 6268637952000, 52260618977280, 994393803303936000, 61479419904000, 40320

Offset: 1

N. J. A. Sloane, Oct 23 2014

By definition, this is the number of nXnXnX...Xn = n^(d+1) arrays of 0's and 1's with exactly one 1 in each row, column, ..., line, ... .
An ordinary permutation is the case d = 1 (ordinary matrices with a single 1 in each row and column).
Rows d=2,3,... correspond to Latin squares, cubes, etc.

			The array begins:
d\n: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
-----------------------------------------------------------
1: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, ...
2: 1, 2, 12, 576, 161280, 812851200, 61479419904000, 108776032459082956800,...
3: 1, 2, 24, 55296, 2781803520, 994393803303936000, ...
4: 1, 2, 48, 36972288, 52260618977280, ...
5: 1, 2, 96, 6268637952000, 2010196727432478720, ...
6: 1, 2, 192, ...
7: 1, 2, 384, ...
8: 1, 2, 768, ...
...

Linial, Nathan, and Zur Luria, An upper bound on the number of high-dimensional permutations, arXiv preprint arXiv:1106.0649 [math.CO], (2011).
Linial, Nathan, and Zur Luria, An upper bound on the number of high-dimensional permutations, Combinatorica, 34 (2014), 471-486.

Rows: A000142, A002860, A098679, A100540, A132206.
Column 4 = A249028.
See A249026 for another version.

cp's OEIS Frontend

A100540 Total number of Latin 4-dimensional hypercubes (Latin polyhedra) of order n.

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A249026 Array read by antidiagonals upwards: T(d,n) = number of d-dimensional permutations of n letters (d >= 0, n >= 1).

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A132205 Number of reduced Latin 5-dimensional hypercubes (Latin polyhedra) of order n.

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A211215 Total number of Latin n-dimensional hypercubes of order 4; labeled n-ary quasigroups of order 4.

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A249027 Array read by antidiagonals upwards: T(d,n) = number of d-dimensional permutations of n letters (d >= 1, n >= 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs