1,2

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.

W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, preprint Oct 2008.

W. Lang, First 10 rows of the array and more.

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).

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 labelled forests comes therefore in 4 versions due to the two ternary root vertices.

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

Sequence in context: A144279 A144280 A107717 this_sequence A000369 A136236 A113090

Adjacent sequences: A143170 A143171 A143172 this_sequence A143174 A143175 A143176

nonn,easy,tabf

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Oct 09 2008