OFFSET
0,3
COMMENTS
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
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
KEYWORD
nonn,tabf,easy
AUTHOR
EXTENSIONS
More terms from Joshua Zucker, Jul 27 2006
a(0)=1 prepended by Andrew Howroyd, Oct 18 2020
STATUS
approved