

A144267


Partition number array, called M32(4), related to A011801(n,m)= S2(4;n,m) ( generalized Stirling triangle).


5



1, 4, 1, 36, 12, 1, 504, 144, 48, 24, 1, 9576, 2520, 1440, 360, 240, 40, 1, 229824, 57456, 30240, 12960, 7560, 8640, 960, 720, 720, 60, 1, 6664896, 1608768, 804384, 635040, 201096, 211680, 90720, 60480, 17640, 30240, 6720, 1260, 1680, 84, 1, 226606464, 53319168
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OFFSET

1,2


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(4;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 (r+3)ary trees if the outdegree is r >= 0.
If M32(4;n,k) is summed over those k with fixed number of parts m one obtains triangle A011801(n,m)= S2(4;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..46.
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(4,j,1)^e(n,k,j),j=1..n) = M3(n,k)*product(S2(4,j,1)^e(n,k,j),j=1..n), with S2(4,n,1)= A008546(n1) = (5*n6)(!^5) (5factorials) 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)=48. The relevant partition of 4 is (2^2). The 48 unordered (0,2,0,0)forests are composed of the following 2 rooted increasing trees 12,34; 13,24 and 14,23. The trees are quaternary because r=1 vertices are quaternary (4ary) and for the leaves (r=0) the arity does not matter. Each of the three differently labeled forests comes therefore in 4^2=16 versions due to the two quaternary root vertices.


CROSSREFS

Cf. A143173 (M32(3) array), A144268 (M32(5) array).
Sequence in context: A061036 A217020 A329066 * A011801 A169656 A303987
Adjacent sequences: A144264 A144265 A144266 * A144268 A144269 A144270


KEYWORD

nonn,easy,tabf


AUTHOR

Wolfdieter Lang, Oct 09 2008


STATUS

approved



