OFFSET
1,4
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.
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.
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).
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Wolfdieter Lang, Oct 09 2008, Dec 04 2008
STATUS
approved