

A143171


Partition number array, called M32(1), related to A001497(n1,m1)= S2(1;n,m) ( generalized Stirling2 triangle).


4



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
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OFFSET

1,4


COMMENTS

Each partition of n, ordered as in AbramowitzStegun (ASt order; for the reference see A134278), is mapped to a nonnegative integer a(n,k)=:M32(1;n,k) with the kth partition of n in ASt 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 kth partition of n in the ASt order. The kth 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 rary trees if the outdegree is r>=0.
This generalizes the array of multinomials called M_3 in AbramowitzStegun, pp. 8312. M_3 = A036040.
If M32(1;n,k) is summed over those k with fixed number of parts m one obtains triangle A001497(n1,m1)= 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.


LINKS

Table of n, a(n) for n=1..56.
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.


FORMULA

a(n,k)= (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(S2(1,j,1)^e(n,k,j),j=1..n) = M3(n,k)*product(S2(1,j,1)^e(n,k,j),j=1..n), with S2(1,n,1)= A001147(n1) = (2*n3)(!^2) (2factorials) for n>=2 and 1 if n=1 and the exponent e(n,k,j) of j in the kth partition of n in the ASt 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 12,34; 13,24 and 14,23. The trees are unary because r=1 vertices are unary (1ary) and for the leaves (r=0) the arity does not matter.


CROSSREFS

A143173 M32(2) array.
Sequence in context: A227343 A216294 A039797 * A112292 A001497 A123244
Adjacent sequences: A143168 A143169 A143170 * A143172 A143173 A143174


KEYWORD

nonn,easy,tabf


AUTHOR

Wolfdieter Lang Oct 09 2008, Dec 04 2008


STATUS

approved



