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A078760 Combinations of a partition: number of ways to label a partition (of size n) with numbers 1 to n. 6
1, 1, 1, 2, 1, 3, 6, 1, 4, 6, 12, 24, 1, 5, 10, 20, 30, 60, 120, 1, 6, 15, 30, 20, 60, 120, 90, 180, 360, 720, 1, 7, 21, 42, 35, 105, 210, 140, 210, 420, 840, 630, 1260, 2520, 5040, 1, 8, 28, 56, 56, 168, 336, 70, 280, 420, 840, 1680, 560, 1120, 1680, 3360, 6720, 2520 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

This is a function of the individual partitions of an integer. The number of values in each line is given by A000041; thus lines 0 to 5 of the sequence are (1), (1), (1,2), (1,3,6), (1,4,6,12,24). The partitions in each line are ordered with the largest part sizes first, so the line 4 indices are [4], [3,1], [2,2], [2,1,1] and [1,1,1,1]. Note that exponents are often used to represent repeated values in a partition, so the last index could instead be written [1^4]. The combination function (sequence A007318) C(n,m) = C([m,n-m]).

This sequence is also the sequence of multinomial coefficients for partitions ordered lexicographically, matching partition sequence A080577. This is different ordering than in sequence A036038 of multinomial coefficients. - Sergei Viznyuk, Mar 15 2012

LINKS

T. D. Noe, Rows n=0..25 of triangle, flattened

Sergei Viznyuk, C Program

Index entries for triangles and arrays related to Pascal's triangle.

FORMULA

C([<a(i)>]) = (Sum a(i))! / Product a(i) !

EXAMPLE

The irregular table starts:

[0] {1},

[1] {1},

[2] {1, 2},

[3] {1, 3,  6},

[4] {1, 4,  6, 12, 24},

[5] {1, 5, 10, 20, 30, 60, 120},

[6] {1, 6, 15, 30, 20, 60, 120, 90, 180, 360, 720}

.

C([2,1]) = 3 for the labelings ({1,2},{3}), ({1,3},{2}) and ({2,3},{2}).

MATHEMATICA

Flatten[Table[Apply[Multinomial, IntegerPartitions[i], {1}], {i, 0, 25}] (* T. D. Noe, Oct 14 2007 *)

Flatten[ Multinomial @@@ IntegerPartitions @ # & /@ Range[ 0, 8]] (* Michael Somos, Feb 05 2011 *)

CROSSREFS

Different from A036038.

Cf. A080577, A000041.

Sequence in context: A171999 A036038 A210237 * A103280 A046899 A309220

Adjacent sequences:  A078757 A078758 A078759 * A078761 A078762 A078763

KEYWORD

nice,easy,nonn,tabf,look

AUTHOR

Franklin T. Adams-Watters, Jan 08 2003

STATUS

approved

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Last modified February 19 19:21 EST 2020. Contains 332047 sequences. (Running on oeis4.)