OFFSET
0,6
COMMENTS
The sum of row n equals A011782(n). The first and last columns equal 1. The number of integers per row equals the partition number p(n). Row n is a vector of weights or multiplicities relating counts of ordered versus unordered objects classified according to the partitions of n.
a(n) is the multinomial coefficient of the signature of the n-th partition. - Franklin T. Adams-Watters, Apr 08 2008
Let f(x)=1/(1-sum(j>=1, c[j]*x^j))=sum(n>=0, w(n)*x^n), then the coefficients of wn=Pn(c[1],...,c[n]), listed in reverse lexicographic order, give row n of T(n,k). - Groux Roland, Mar 08 2011
LINKS
Alois P. Heinz, Rows n = 0..26, flattened
EXAMPLE
The partitions of 4 are {4}, {3,1}, {2,2}, {2,1,1}, {1,1,1,1}, so the fourth row equals 1,2,1,3,1 since these are the counts of the permutations of these lists.
Triangle begins:
1;
1;
1, 1;
1, 2, 1;
1, 2, 1, 3, 1;
1, 2, 2, 3, 3, 4, 1;
1, 2, 2, 3, 1, 6, 4, 1, 6, 5, 1;
MATHEMATICA
mult[li:{__Integer}] := Apply[Multinomial, Length/@Split[ Sort[li] ] ]; Table[mult/@Partitions[n], {n, 12}]
PROG
(PARI) \\ here mulp(v) computes the multiplicity of the given partition.
mulp(v) = {my(p=(#v)!, k=1); for(i=2, #v, k=if(v[i]==v[i-1], k+1, p/=k!; 1)); p/k!}
Row(n)={apply(mulp, vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{ for(n=0, 9, print(Row(n))) } \\ Peter Dolland, Nov 11 2019
CROSSREFS
KEYWORD
AUTHOR
Wouter Meeussen, Aug 09 2002
STATUS
approved