A119270 -id:A119270

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A119269

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

+10
9

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,8

COMMENTS

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

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

LINKS

Table of n, a(n) for n=1..91.

FORMULA

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

EXAMPLE

Table starts:

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

CROSSREFS

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

Cf. A119339, A119340, A119341, A119342.

KEYWORD

nonn,tabl

AUTHOR

Franklin T. Adams-Watters, May 11 2006

EXTENSIONS

More terms from Max Alekseyev, May 14 2006

STATUS

approved

A119338

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

+10
9

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,9

COMMENTS

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

LINKS

Table of n, a(n) for n=1..91.

EXAMPLE

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, ...

...

CROSSREFS

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

KEYWORD

nonn,tabl

AUTHOR

Max Alekseyev, May 15 2006

STATUS

approved

A118364

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

+10
5

0, 1, 2, 6, 19, 60

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Partial sums are given by A118365.

LINKS

Table of n, a(n) for n=1..6.

FORMULA

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

CROSSREFS

Cf. A119270, A119339, A119268.

KEYWORD

hard,more,nonn

AUTHOR

Max Alekseyev, May 16 2006

STATUS

approved

A119339

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

+10
4

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

LINKS

Table of n, a(n) for n=0..84.

EXAMPLE

Table starts:

1,0

1,1,0

1,2,1,0

1,2,3,1,0

CROSSREFS

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

KEYWORD

nonn,tabl

AUTHOR

Max Alekseyev, May 15 2006

STATUS

approved

A119271

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

+10
3

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,9

COMMENTS

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

LINKS

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, 2012.

FORMULA

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.

EXAMPLE

Table starts:

0,1,

0,1,1,

0,1,3,1,

0,1,5,6,1,

CROSSREFS

Cf. A119270, A096806. Column 1 is A007042.

KEYWORD

nonn,tabl,hard

AUTHOR

Franklin T. Adams-Watters, May 11 2006

STATUS

approved

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