

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
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OFFSET

1,2


COMMENTS

Each partition of n, ordered as in AbramowitzStegun (ASt order; for the reference see A134278), is mapped to a nonnegative integer a(n,k)=:M32(2;n,k) with the kth partition of n in ASt 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 kth partition of n in the ASt order. The kth 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

Table of n, a(n) for n=1..50.
W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.


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(n1) = (3*n4)(!^3) (3factorials) for n>=2 and 1 if n=1 and the exponent e(n,k,j) of j in the kth partition of n in the ASt 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 12,34; 13,24 and 14,23. The trees are binary because r=1 vertices are binary (2ary) and for the leaves (r=0) the arity does not matter. Each of the three differently labelled forests comes therefore in 4 versions due to the two binary root vertices.


CROSSREFS

A143171 (M32(1) array), A143173 (M32(3) array).
Sequence in context: A112333 A066868 A193900 * A004747 A155810 A225470
Adjacent sequences: A143169 A143170 A143171 * A143173 A143174 A143175


KEYWORD

nonn,easy,tabf


AUTHOR

Wolfdieter Lang Oct 09 2008


STATUS

approved



