A143171 Partition number array, called M32(-1), related to A001497(n-1,m-1) = |S2(-1;n,m)| (generalized Stirling2 triangle).

1, 1, 1, 3, 3, 1, 15, 12, 3, 6, 1, 105, 75, 30, 30, 15, 10, 1, 945, 630, 225, 90, 225, 180, 15, 60, 45, 15, 1, 10395, 6615, 2205, 1575, 2205, 1575, 630, 315, 525, 630, 105, 105, 105, 21, 1, 135135, 83160, 26460, 17640, 7875, 26460, 17640, 12600, 3150, 2520, 5880, 6300

Offset: 1

Author

Wolfdieter Lang, Oct 09 2008, Dec 04 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(-1;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_{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.

This generalizes the array of multinomials called M_3 in Abramowitz-Stegun, pp. 831-2. M_3 = A036040.

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.

Examples

			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.
		

Links

• Wolfdieter Lang, First 10 rows of the array and more.
• Wolfdieter Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Crossrefs

Cf. A143173 M32(-2) array.

Formula

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).