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A072811
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T(n,k)=multiplicity of the k-th partition of n in Mathematica order, defined to be the count of its permutations (compositions).
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2
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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; internal format)
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OFFSET
| 0,6
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COMMENTS
| The sum of row n equals 2^(n-1). 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. rows: 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
a(n) is the multinomial coefficient of the signature of the n-th partition. - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), 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.
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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.
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MATHEMATICA
| mult[li:{__Integer}] := Apply[Multinomial, Length/@Split[ Sort[li] ] ]; Table[mult/@Partitions[n], {n, 12}]
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CROSSREFS
| Cf. A080577, A080575, A115621, A102462.
Sequence in context: A165357 A048996 A111786 * A080027 A050305 A117164
Adjacent sequences: A072808 A072809 A072810 * A072812 A072813 A072814
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KEYWORD
| easy,nonn,tabf
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AUTHOR
| Wouter Meeussen (wouter.meeussen(AT)pandora.be), Aug 09 2002
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