A035206 - OEIS

A035206

Number of multisets associated with least integer of each prime signature.

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

0,3

COMMENTS

a(n,k) multiplied by A036038(n,k) yields A049009(n,k).

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]

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)

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].

Wolfdieter Lang, First 10 rows and more.

FORMULA

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.

EXAMPLE

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

[rewritten and extended table by Wolfdieter Lang, Jul 11 2012]

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

PROG

(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)])}

{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 18 2020

CROSSREFS

Cf. A036038, A048996, A049009.

Cf. A001700 (row sums).

Cf. A103371(n-1, m-1) (triangle obtained after summing in every row the numbers with like part numbers m).

Sequence in context: A046899 A309220 A225632 * A210238 A209936 A213941

Adjacent sequences: A035203 A035204 A035205 * A035207 A035208 A035209