A119271 -id:A119271 - cp's OEIS frontend

A096806 Triangle, read by rows, such that the binomial transform of the n-th row lists the m-dimensional partitions of n, for n>=1 and m>=0.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 11, 7, 1, 1, 10, 27, 28, 11, 1, 1, 14, 57, 93, 64, 16, 1, 1, 21, 117, 269, 282, 131, 22, 1, 1, 29, 223, 707, 1062, 766, 244, 29, 1, 1, 41, 417, 1747, 3565, 3681, 1871, 421, 37, 1, 1, 55, 748, 4090, 10999, 15489, 11400, 4152, 683, 46, 1, 1, 76

Offset: 1

Paul D. Hanna, Jul 19 2004

The n-th row equals the inverse binomial transform of n-th column of square array A096751, for n>=1. The zero-dimensional partition of n is taken to be 1 for all n.

			The number of m-dimensional partitions of 5, for m>=0, is given by the binomial transform of the 5th row:
BINOMIAL([1,6,11,7,1]) = [1,7,24,59,120,216,357,554,819,1165,...] = A008779.
Rows begin:
  [1],
  [1,  1],
  [1,  2,   1],
  [1,  4,   4,    1],
  [1,  6,  11,    7,     1],
  [1, 10,  27,   28,    11,     1],
  [1, 14,  57,   93,    64,    16,      1],
  [1, 21, 117,  269,   282,   131,     22,      1],
  [1, 29, 223,  707,  1062,   766,    244,     29,     1],
  [1, 41, 417, 1747,  3565,  3681,   1871,    421,    37,     1],
  [1, 55, 748, 4090, 10999, 15489,  11400,   4152,   683,    46,    1],
  [1, 76,1326, 9219, 31828, 58975,  59433,  31802,  8483,  1054,   56,   1],
  [1,100,2284,20095, 87490,207735, 276230, 204072, 80664, 16162, 1561,  67, 1],
  [1,134,3898,42707,230737,687665,1173533,1148939,632478,188077,29031,2234,79,1],
  ...
The inverse binomial transform of the diagonals of this triangle begin:
  [1],
  [1, 1,  1],
  [1, 3,  4,   6,  3],
  [1, 5, 16,  29,  49,   45,   15],
  [1, 9, 38, 127, 289,  540,  660,   420, 105],
  [1,13, 90, 397,1384, 3633, 7506, 10920,9765,4725,945],
  [1,20,182,1140,5266,19324,55645,125447,  ? ,  ? , ?  ,62370,10395],
  ...

S. Govindarajan, Notes on higher-dimensional partitions, arXiv:1203.4419 [math.CO], 2012.

Cf. A096751, A096807 (row sums), A000065 (column k=1?), A008778 (bin trans 4th row), A042984 (bin trans 6th row)

Cf. A119271.

T(n, 0)=T(n, n-1)=1, T(n, 1)=A000041(n)-1, T(n, n-2)=(n-1)*(n-2)/2+1, for n>=1.

A119270 Triangle: number of exactly (m-1)-dimensional partitions of n, up to conjugacy, for n >= 1, m >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 1, 5, 5, 2, 1, 0, 1, 7, 11, 6, 2, 1, 0, 1, 11, 21, 16, 6, 2, 1, 0, 1, 15, 39, 38, 18, 6, 2, 1, 0, 1, 21, 73, 86, 51, 19, 6, 2, 1, 0, 1, 28, 129, 193, 135, 57, 19, 6, 2, 1, 0, 1, 39, 227, 420, 352, 170, 59, 19, 6, 2, 1, 0, 1, 51, 390, 890, 894

Offset: 1

Franklin T. Adams-Watters, May 11 2006

The partition of 1 is considered to be dimension -1 by convention.

Partitions are considered as generalized Ferrers diagrams; any permutation of the axes produces a conjugate.

			Table starts:
1
0,1
0,1,1
0,1,2,1
0,1,3,2,1

Cf. A119269, A119271.

Reversed triangle is A119339. Columns stabilize to A118364.

a(n,m) = A119269(n,m)-A119269(n,m-1).

More terms from Max Alekseyev, May 15 2006

cp's OEIS Frontend

A096806 Triangle, read by rows, such that the binomial transform of the n-th row lists the m-dimensional partitions of n, for n>=1 and m>=0.

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