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A072811 T(n,k) = multiplicity of the k-th partition of n in Mathematica order, defined to be the count of its permutations (compositions). 4
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, 1, 2, 2, 3, 2, 6, 4, 3, 3, 12, 5, 4, 10, 6, 1, 1, 2, 2, 3, 2, 6, 4, 1, 6, 3, 12, 5, 3, 6, 12, 20, 6, 1, 10, 15, 7, 1, 1, 2, 2, 3, 2, 6, 4, 2, 6, 3, 12, 5, 3, 6, 12, 12, 20, 6, 1, 12, 10, 4, 30, 30, 7, 5, 20, 21, 8, 1 (list; graph; refs; listen; history; text; internal format)
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

Cf. A080577, A080575, A115621, A102462.

Sequence in context: A250007 A048996 A111786 * A296559 A233548 A080027

Adjacent sequences:  A072808 A072809 A072810 * A072812 A072813 A072814

KEYWORD

easy,nonn,look,tabf,changed

AUTHOR

Wouter Meeussen, Aug 09 2002

STATUS

approved

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Last modified November 17 08:39 EST 2019. Contains 329217 sequences. (Running on oeis4.)