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A035206
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Number of multisets associated with least integer of each prime signature.
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15
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1, 1, 2, 1, 3, 6, 1, 4, 12, 6, 12, 1, 5, 20, 20, 30, 30, 20, 1, 6, 30, 30, 15, 60, 120, 20, 60, 90, 30, 1, 7, 42, 42, 42, 105, 210, 105, 105, 140, 420, 140, 105, 210, 42, 1, 8, 56, 56, 56, 28, 168, 336, 336, 168, 168, 280, 840, 420, 840, 70, 280, 1120, 560, 168, 420, 56, 1, 9, 72
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OFFSET
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0,3
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COMMENTS
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a(n,k) enumerates distributions of n identical objects (balls) into m of altogether n distinguishable boxes. The k-th partition of n, taken in the Abramowitz-Stegun (A-St) order, specifies the occupation of the m =m(n,k)= A036043(n,k) boxes. m = m(n,k) is the number of parts of the k-th partition of n. For the A-St ordering see pp.831-2 of the reference given in A117506. - Wolfdieter Lang, Nov 13 2007.
The sequence of row lengths is p(n)= A000041(n) (partition numbers).
For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
The corresponding triangle with summed row entries which belong to partitions of the same number of parts k is A103371. [Wolfdieter Lang, Jul 11 2012]
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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a(n,k) = A048996(n,k)*binomial(n,m(n,k)),n>=1, k=1,...,p(n) and m(n,k):=A036043(n,k) gives the number of parts of the k-th partition of n.
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EXAMPLE
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n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 1
1 1
2 2 1
3 3 6 1
4 4 12 6 12 1
5 5 20 20 30 30 20 1
6 6 30 30 15 60 120 20 60 90 30 1
7 7 42 42 42 105 210 105 105 140 420 140 105 210 42 1
...
Row No. 8: 8 56 56 56 28 168 336 336 168 168 280 840 420 840 70 280 1120 560 168 420 56 1
Row No. 9: 9 72 72 72 72 252 504 504 252 252 504 84 504 1512 1512 1512 1512 504 630 2520 1260 3780 630 504 2520 1680 252 756 72 1
a(5,5) relates to the partition (1,2^2) of n=5. Here m=3 and 5 indistinguishable (identical) balls are put into boxes b1,...,b5 with m=3 boxes occupied; one with one ball and two with two balls.
Therefore a(5,5) = binomial(5,3)*3!/(1!*2!) = 10*3 = 30. Wolfdieter Lang, Nov 13 2007.
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PROG
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(PARI)
C(sig)={my(S=Set(sig)); binomial(vecsum(sig), #sig)*(#sig)!/prod(k=1, #S, (#select(t->t==S[k], sig))!)}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
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CROSSREFS
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Cf. A103371(n-1, m-1) (triangle obtained after summing in every row the numbers with like part numbers m).
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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