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%I #17 Jul 02 2023 02:01:21
%S 1,1,1,3,3,1,15,12,3,6,1,105,75,30,30,15,10,1,945,630,225,90,225,180,
%T 15,60,45,15,1,10395,6615,2205,1575,2205,1575,630,315,525,630,105,105,
%U 105,21,1,135135,83160,26460,17640,7875,26460,17640,12600,3150,2520,5880,6300
%N Partition number array, called M32(-1), related to A001497(n-1,m-1) = |S2(-1;n,m)| (generalized Stirling2 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(-1;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_{j=1..n} e(n,k,j). The special (enk)-forest is composed of m rooted increasing r-ary trees if the outdegree is r >= 0.
%C This generalizes the array of multinomials called M_3 in Abramowitz-Stegun, pp. 831-2. M_3 = A036040.
%C If M32(-1;n,k) is summed over those k with fixed number of parts m one obtains triangle A001497(n-1,m-1) = |S2(-1;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 Wolfdieter Lang, <a href="/A143171/a143171.txt">First 10 rows of the array and more</a>.
%H Wolfdieter 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_{j=1..n} e(n,k,j)!*j!^e(n,k,j)) * Product_{j=1..n} |S2(-1,j,1)|^e(n,k,j) = M3(n,k)*Product_{j=1..n} |S2(-1,j,1)|^e(n,k,j), with |S2(-1,n,1)| = A001147(n-1) = (2*n-3)(!^2) (2-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) = 3. The relevant partition of 4 is (2^2). The 3 unordered (0,2,0,0)-forests are composed of the following 2 rooted increasing unary trees 1--2,3--4; 1--3,2--4 and 1--4,2--3. The trees are unary because r=1 vertices are unary (1-ary) and for the leaves (r=0) the arity does not matter.
%Y Cf. A143173 M32(-2) array.
%K nonn,easy,tabf
%O 1,4
%A _Wolfdieter Lang_, Oct 09 2008, Dec 04 2008
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