A119270 -id:A119270 - cp's OEIS frontend

A119269 Table by antidiagonals: number of m-dimensional partitions of n up to conjugacy, for n >= 1, m >= 0.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 4, 2, 1, 1, 1, 6, 6, 4, 2, 1, 1, 1, 8, 11, 7, 4, 2, 1, 1, 1, 12, 19, 13, 7, 4, 2, 1, 1, 1, 16, 33, 25, 14, 7, 4, 2, 1, 1, 1, 22, 55, 49, 27, 14, 7, 4, 2, 1, 1, 1, 29, 95, 93, 55, 28, 14, 7, 4, 2, 1, 1, 1, 40, 158, 181, 111, 57, 28, 14, 7, 4, 2, 1, 1

Offset: 1

Franklin T. Adams-Watters, May 11 2006

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

Transposed table is A119338. - Max Alekseyev, May 14 2006

			Table starts:
  1, 1,  1,  1,  1
  1, 1,  1,  1,  1
  1, 2,  2,  2,  2
  1, 3,  4,  4,  4
  1, 4,  6,  7,  7
  1, 6, 11, 13, 14

Columns A005987, A000786, A119266, A119267; diagonal A119268. Cf. A096751, A119270.

Cf. A119339, A119340, A119341, A119342.

a(n,m) = a(n,n-2) for m >= n-1.

More terms from Max Alekseyev, May 14 2006

A119338 Table by antidiagonals: a(m,n) is the number of m-dimensional partitions of n up to conjugacy, for m >= 0, n >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 4, 4, 1, 1, 1, 2, 4, 6, 6, 1, 1, 1, 2, 4, 7, 11, 8, 1, 1, 1, 2, 4, 7, 13, 19, 12, 1, 1, 1, 2, 4, 7, 14, 25, 33, 16, 1, 1, 1, 2, 4, 7, 14, 27, 49, 55, 22, 1, 1, 1, 2, 4, 7, 14, 28, 55, 93, 95, 29, 1, 1, 1, 2, 4, 7, 14, 28, 57, 111, 181, 158, 40, 1

Offset: 1

Max Alekseyev, May 15 2006

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

			Table starts:
  1, 1, 1, 1, 1,  1, ...
  1, 1, 2, 3, 4,  6, ...
  1, 1, 2, 4, 6, 11, ...
  1, 1, 2, 4, 7, 13, ...
  1, 1, 2, 4, 7, 14, ...
  ...

Rows: A000012, A046682, A000786, A119266, A119267, A119340, A119341, A119342 stabilize to A119268. Transposed table is A119269. Cf. A119339, A119270, A118364, A118365.

A118364 Limit of the number of exactly m-dimensional partitions of m+n as m tends to infinity.

Original entry on oeis.org

0, 1, 2, 6, 19, 60

Offset: 1

Max Alekseyev, May 16 2006

Partial sums are given by A118365.

Cf. A119270, A119339, A119268.

a(n)=A119270(m+n,m)=A119339(m+n,n) for all m>=2n-5

A119339 Triangle: number of exactly (m-1)-dimensional partitions of n, up to conjugacy, for n >= 1, m=n..1.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 3, 1, 0, 1, 2, 5, 5, 1, 0, 1, 2, 6, 11, 7, 1, 0, 1, 2, 6, 16, 21, 11, 1, 0, 1, 2, 6, 18, 38, 39, 15, 1, 0, 1, 2, 6, 19, 51, 86, 73, 21, 1, 0, 1, 2, 6, 19, 57, 135, 193, 129, 28, 1, 0, 1, 2, 6, 19, 59, 170, 352, 420, 227, 39, 1, 0, 1, 2, 6, 19, 60, 186, 498

Offset: 0

Max Alekseyev, May 15 2006

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

Reversed triangle is A119270. Diagonals stabilize to A118364. Cf. A119269, A119338.

A119271 Triangle: number of exactly (m-1)-dimensional partitions of n, for n >= 1, m >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 5, 6, 1, 0, 1, 9, 18, 10, 1, 0, 1, 13, 44, 49, 15, 1, 0, 1, 20, 97, 172, 110, 21, 1, 0, 1, 28, 195, 512, 550, 216, 28, 1, 0, 1, 40, 377, 1370, 2195, 1486, 385, 36, 1, 0, 1, 54, 694, 3396, 7603, 7886, 3514, 638, 45, 1, 0, 1, 75, 1251, 7968

Offset: 1

Franklin T. Adams-Watters, May 11 2006

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

			Table starts:
  1,
  0,1,
  0,1,1,
  0,1,3,1,
  0,1,5,6,1,
  ...

Suresh Govindarajan, Rows n = 1..26 of Triangle
Suresh Govindarajan, Partitions Generator (gives partitions of integers <= 25 in any dimension using this triangle).
Suresh Govindarajan, Refined counting of higher-dimensional partitions
Suresh Govindarajan, Notes on higher-dimensional partitions, arXiv preprint arXiv:1203.4419 [math.CO], 2012.

Cf. A119270, A096806. Column 1 is A007042.

a(n,m) = A096806(n,m-1)-a(n,m-1). Binomial transform of n-th row lists the (m-1) dimensional partitions of n.

cp's OEIS Frontend

A119269 Table by antidiagonals: number of m-dimensional partitions of n up to conjugacy, for n >= 1, m >= 0.

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A119338 Table by antidiagonals: a(m,n) is the number of m-dimensional partitions of n up to conjugacy, for m >= 0, n >= 1.

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A118364 Limit of the number of exactly m-dimensional partitions of m+n as m tends to infinity.

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A119339 Triangle: number of exactly (m-1)-dimensional partitions of n, up to conjugacy, for n >= 1, m=n..1.

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A119271 Triangle: number of exactly (m-1)-dimensional partitions of n, for n >= 1, m >= 0.

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