A143173 Partition number array, called M32(-3), related to A000369(n,m) = |S2(-3;n,m)| (generalized Stirling triangle).

1, 3, 1, 21, 9, 1, 231, 84, 27, 18, 1, 3465, 1155, 630, 210, 135, 30, 1, 65835, 20790, 10395, 4410, 3465, 3780, 405, 420, 405, 45, 1, 1514205, 460845, 218295, 169785, 72765, 72765, 30870, 19845, 8085, 13230, 2835, 735, 945, 63, 1, 40883535, 12113640, 5530140, 4074840

Offset: 1

Author

Wolfdieter Lang, Oct 09 2008

Keywords

• nonn
• easy
• tabf

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(-3;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+2)-ary trees if the outdegree is r >= 0.

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.

Examples

			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 1--2,3--4; 1--3,2--4 and 1--4,2--3. The trees are ternary because r=1 vertices are ternary (3-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 ternary root vertices.
		

Links

• 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.

Crossrefs

Cf. A143172 (M32(-2) array), A144267 (M32(-4) array).

Formula

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(n-1) = (4*n-5)(!^4) (4-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).