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

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

0,3

Comments

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.

Links

Table of n, a(n) for n=0..44.

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.

Example

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, ...
...

Crossrefs

Rows: A000142, A002860, A098679, A100540, A132206.

Column 4 = A249028.

See A249027 for another version.

Sequence in context: A126247 A213999 A374411 * A263597 A263905 A263693

Adjacent sequences: A249023 A249024 A249025 * A249027 A249028 A249029

Keyword

nonn,tabl

Author

N. J. A. Sloane, Oct 23 2014

Status

approved