|
|
A143172
|
|
Partition number array, called M32(-2), related to A004747(n,m) = |S2(-2;n,m)| (generalized Stirling triangle).
|
|
4
|
|
|
1, 2, 1, 10, 6, 1, 80, 40, 12, 12, 1, 880, 400, 200, 100, 60, 20, 1, 12320, 5280, 2400, 1000, 1200, 1200, 120, 200, 180, 30, 1, 209440, 86240, 36960, 28000, 18480, 16800, 7000, 4200, 2800, 4200, 840, 350, 420, 42, 1, 4188800, 1675520, 689920, 492800, 224000, 344960
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k)=:M32(-2;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
a(n,k) enumerates special unordered forests related to the k-th partition of n in the A-St order. The k-th partition of n is given by the exponents enk =(e(n,k,1),...,e(n,k,n)) of 1,2,...n. The number of parts is m = sum(e(n,k,j),j=1..n). The special (enk)-forest is composed of m rooted increasing (r+1)-ary trees if the outdegree is r>=0.
If M32(-2;n,k) is summed over those k with fixed number of parts m one obtains triangle A004747(n,m)= |S2(-2;n,m)|, a generalization of Stirling numbers of the second kind. For S2(K;n,m), K from the integers, see the reference under A035342.
|
|
LINKS
|
|
|
FORMULA
|
a(n,k)= (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S2(-2,j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S2(-2,j,1)|^e(n,k,j),j=1..n), with |S2(-2,n,1)|= A008544(n-1) = (3*n-4)(!^3) (3-factorials) for n>=2 and 1 if n=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1. M3(n,k):= A036040(n,k), k=1..p(n), p(n):= A000041(n).
|
|
EXAMPLE
|
a(4,3)=12. The relevant partition of 4 is (2^2). The 12 unordered (0,2,0,0)-forests are composed of the following 2 rooted increasing trees 1--2,3--4; 1--3,2--4 and 1--4,2--3. The trees are binary because r=1 vertices are binary (2-ary) and for the leaves (r=0) the arity does not matter. Each of the three differently labeled forests comes therefore in 4 versions due to the two binary root vertices.
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy,tabf
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|