A066991 Square array read by descending antidiagonals of number of ways of dividing n*k labeled items into k unlabeled orders with n items in each order.

1, 1, 2, 1, 12, 6, 1, 120, 360, 24, 1, 1680, 60480, 20160, 120, 1, 30240, 19958400, 79833600, 1814400, 720, 1, 665280, 10897286400, 871782912000, 217945728000, 239500800, 5040, 1, 17297280, 8892185702400, 20274183401472000

Offset

1,3

Comments

T(p,k) = (pk)!/k! is divisible by p^k but not p^(k+1) for p prime; e.g., T(3,4) = 3^4*11*10*8*7*5*4*2*1 = 19958400.

Links

Paolo Xausa, Table of n, a(n) for n = 1..820 (antidiagonals 1..40 of the array, flattened).

Formula

T(n,k) = (n*k)!/k!.

Example

The array begins:
  n\k|   1        2             3                   4  ...
  --------------------------------------------------------
   1 |   1,       1,            1,                  1, ...
   2 |   2,      12,          120,               1680, ...
   3 |   6,     360,        60480,           19958400, ...
   4 |  24,   20160,     79833600,       871782912000, ...
   5 | 120, 1814400, 217945728000, 101370917007360000, ...
  ...

Mathematica

Table[((n-k+1)*k)!/k!, {n, 10}, {k, n, 1, -1}] (* Paolo Xausa, Feb 19 2024 *)

Crossrefs

Rows include A000012, A001813, A064350.

Columns include A000142, A002674, A065961.

Cf. A060538, A060539, A060540.

Sequence in context: A342587 A008285 A119274 * A132875 A050139 A010255

Adjacent sequences: A066988 A066989 A066990 * A066992 A066993 A066994

Keyword

nonn,tabl

Author

Henry Bottomley, Feb 01 2002

Extensions

Edited by Paolo Xausa, Feb 19 2024

Status

approved