A330490 Total number of permutation arrays of side length n and dimension k as defined by Eriksson and Linusson (2000a); square array T(n,k), read by ascending antidiagonals, for n, k >= 1.

1, 1, 1, 1, 2, 1, 1, 6, 5, 1, 1, 24, 70, 15, 1, 1, 120, 2167, 1574, 52, 1, 1, 720, 130708, 968162, 69874, 203, 1, 1, 5040, 14231289

Offset: 1

Author

Petros Hadjicostas, Dec 16 2019

Keywords

• nonn
• tabl
• more

Comments

The poset P_{3 x 3} of (3 x 3 x 3)-permutation arrays is shown in Figure 1 on p. 209 of Eriksson and Linuson (2000a). We have |P_{3 x 3}| = T(3,3) = 70. The numbers in this rectangular array are copied from Table 1 (p. 210) of the same paper.

Examples

			Array T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows, where * indicates a missing number:
  1,     1,            1,      1,     1, ...
  1,     2,            5,     15,    52, ...
  1,     6,           70,   1574, 69874, ...
  1,    24,         2167, 968162,     *, ...
  1,   120,       130708,      *,     *, ...
  1,   720,     14231289,      *,     *, ...
  1,  5040,   2664334184,      *,     *, ...
  1, 40320, 831478035698,      *,     *, ...
  ...
		

Links

• Kimmo Eriksson and Svante Linusson, A combinatorial theory of higher-dimensional permutation arrays, Adv. Appl. Math. 25(2) (2000a), 194-211.
• Kimmo Eriksson and Svante Linusson, A decomposition of Fl(n)^d indexed by permutation arrays, Adv. Appl. Math. 25(2) (2000b), 212-227. [Fl(n)^d denotes the flag manifold over C^n.]

Crossrefs

Cf. A000012, A000110, A000142, A225816.

Formula

T(n=1,k) = 1 = A000012(n) and T(n=2,k) = A000110(k) (Bell numbers).

T(n,k=1) = 1 = A000012(n) and T(n,k=2) = n! = A000142(n).

T(n,k) >= (n!)^(k-1) = A225816(k-1, n).

T(n,k=3) <= n!*2^(binomial(n+1,2) - 1).