

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,nm]).
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. AdamsWatters, Jan 08 2003


STATUS

approved



