%I
%S 1,3,1,21,9,1,231,84,27,18,1,3465,1155,630,210,135,30,1,65835,20790,
%T 10395,4410,3465,3780,405,420,405,45,1,1514205,460845,218295,169785,
%U 72765,72765,30870,19845,8085,13230,2835,735,945,63,1,40883535,12113640,5530140,4074840
%N Partition number array, called M32(3), related to A000369(n,m) = S2(3;n,m) (generalized Stirling triangle).
%C 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(3;n,k) with the kth partition of n in ASt order.
%C The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
%C 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+2)ary trees if the outdegree is r >= 0.
%C If M32(3;n,k) is summed over those k with fixed number of parts m one obtains triangle A000369(n,m) = S2(3;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.
%H W. Lang, <a href="http://www.itp.kit.edu/~wl/EISpub/A143173.text">First 10 rows of the array and more.</a>
%H W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Lang/lang.html">Combinatorial Interpretation of Generalized Stirling Numbers</a>, J. Int. Seqs. Vol. 12 (2009) 09.3.3.
%F a(n,k)= (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(S2(3,j,1)^e(n,k,j),j=1..n) = M3(n,k)*product(S2(3,j,1)^e(n,k,j),j=1..n), with S2(3,n,1)= A008545(n1) = (4*n5)(!^4) (4factorials) 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).
%e a(4,3)=27. 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 ternary because r=1 vertices are ternary (3ary) 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 ternary root vertices.
%Y Cf. A143172 (M32(2) array), A144267 (M32(4) array).
%K nonn,easy,tabf
%O 1,2
%A _Wolfdieter Lang_, Oct 09 2008
