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%I #14 Aug 28 2019 18:01:38
%S 1,4,1,36,12,1,504,144,48,24,1,9576,2520,1440,360,240,40,1,229824,
%T 57456,30240,12960,7560,8640,960,720,720,60,1,6664896,1608768,804384,
%U 635040,201096,211680,90720,60480,17640,30240,6720,1260,1680,84,1,226606464,53319168
%N Partition number array, called M32(-4), related to A011801(n,m)= |S2(-4;n,m)| ( generalized Stirling triangle).
%C 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(-4;n,k) with the k-th partition of n in A-St 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 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+3)-ary trees if the outdegree is r >= 0.
%C If M32(-4;n,k) is summed over those k with fixed number of parts m one obtains triangle A011801(n,m)= |S2(-4;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="/A144267/a144267.txt">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(-4,j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S2(-4,j,1)|^e(n,k,j),j=1..n), with |S2(-4,n,1)|= A008546(n-1) = (5*n-6)(!^5) (5-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).
%e a(4,3)=48. The relevant partition of 4 is (2^2). The 48 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 quaternary because r=1 vertices are quaternary (4-ary) and for the leaves (r=0) the arity does not matter. Each of the three differently labeled forests comes therefore in 4^2=16 versions due to the two quaternary root vertices.
%Y Cf. A143173 (M32(-3) array), A144268 (M32(-5) array).
%K nonn,easy,tabf
%O 1,2
%A _Wolfdieter Lang_, Oct 09 2008
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