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%I #52 Jun 09 2021 05:24:26
%S 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,
%T 180,360,720,1,7,21,42,35,105,210,140,210,420,840,630,1260,2520,5040,
%U 1,8,28,56,56,168,336,70,280,420,840,1680,560,1120,1680,3360,6720,2520
%N Combinations of a partition: number of ways to label a partition (of size n) with numbers 1 to n.
%C 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]).
%C 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
%H T. D. Noe, <a href="/A078760/b078760.txt">Rows n=0..25 of triangle, flattened</a>
%H S.-H. Cha, E. G. DuCasse, and L. V. Quintas, <a href="http://arxiv.org/abs/1405.5283">Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures</a>, arxiv:1405.5283 [math.NT], 2014.
%H Sergei Viznyuk, <a href="http://phystech.com/ftp/s_A209936.c">C Program</a>
%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>.
%F C([<a(i)>]) = (Sum a(i))! / Product a(i) !.
%F T(n,k) = A008480(A063008(n,k)). - _Andrew Howroyd_, Mar 25 2020
%e The irregular table starts:
%e [0] {1},
%e [1] {1},
%e [2] {1, 2},
%e [3] {1, 3, 6},
%e [4] {1, 4, 6, 12, 24},
%e [5] {1, 5, 10, 20, 30, 60, 120},
%e [6] {1, 6, 15, 30, 20, 60, 120, 90, 180, 360, 720}
%e .
%e C([2,1]) = 3 for the labelings ({1,2},{3}), ({1,3},{2}) and ({2,3},{2}).
%p g:= n-> (l-> add(i, i=l)!/mul(i!, i=l))(map(i-> i[2], ifactors(n)[2])):
%p b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
%p [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
%p T:= n-> map(x-> g(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:
%p seq(T(n), n=0..9); # _Alois P. Heinz_, Mar 25 2020
%t Flatten[Table[Apply[Multinomial, IntegerPartitions[i], {1}], {i,0,25}] (* _T. D. Noe_, Oct 14 2007 *)
%t Flatten[ Multinomial @@@ IntegerPartitions @ # & /@ Range[ 0, 8]] (* _Michael Somos_, Feb 05 2011 *)
%t g[n_] := With[{ee = FactorInteger[n][[All, 2]]}, Total[ee]!/Times@@(ee!)];
%t b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {Table[1, {n}]}, Join[ Prepend[#, i] & /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
%t row[n_] := Product[Prime[i]^#[[i]], {i, 1, Length[#]}] & /@ b[n, n];
%t T[n_] := g /@ row[n];
%t T /@ Range[0, 9] // Flatten (* _Jean-François Alcover_, Jun 09 2021, after _Alois P. Heinz_ *)
%o (PARI)
%o C(sig)={vecsum(sig)!/vecprod(apply(k->k!, sig))}
%o Row(n)={apply(C, vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
%o { for(n=0, 8, print(Row(n))) } \\ _Andrew Howroyd_, Mar 25 2020
%Y Different from A036038.
%Y Cf. A000041, A008480, A063008, A080577.
%K nice,easy,nonn,tabf,look
%O 0,4
%A _Franklin T. Adams-Watters_, Jan 08 2003
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