OFFSET
1,2
COMMENTS
For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
Partition number array M_3(4), the k=4 member of a family of generalizations of the multinomial number array M_3 = M_3(1) = A036040.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
The S2(4,n,m) numbers (generalized Stirling2 numbers) are obtained by summing in row n all numbers with the same part number m. In the same manner the S2(n,m) (Stirling2) numbers A008277 are obtained from the partition array M_3= A036040.
a(n,k) enumerates unordered forests of increasing quaternary trees related to the k-th partition of n in the A-St order. The m-forest is composed of m such trees, with m the number of parts of the partition.
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
W. Lang, First 10 rows and more.
FORMULA
EXAMPLE
[1]; [4,1]; [28,12,1]; [280,112,48,24,1]; [3640,1400,1120,280,240,40,1]; ...
a(4,3)=48 from the third (k=3) partition (2^2) of 4: 4!*((4/2!)^2)/2 = 48, because S2(4,2,1) = 4!!! = 4*1 = 4.
There are a(4,3) = 48 = 3*4^2 unordered 2-forests with 4 vertices, composed of two increasing quaternary (4-ary) trees, each with two vertices: there are 3 increasing labelings (1,2)(3,4); (1,3)(2,4); (1,4)(2,3) and each tree comes in four versions from the quaternary structure.
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Wolfdieter Lang, Nov 13 2007
STATUS
approved