1,2

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.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].

W. Lang, First 10 rows and more.

a(n,k)= n!*product((S2(4,j,1)/j!)^e(n,k,j)/e(n,k,j)!,j=1..n) with S2(4,n,1)=A035469(n,1) = A007559(n) = (3*n-2)!!! (triple- or 3-factorials), 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.

[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-forest with 4 vertices, composed of two increasing quaternary (4-ary) trees, each with two vertices: there are 3 increasing labellings (1,2)(3,4); (1,3)(2,4); (1,4)(2,3) and each tree comes in four versions from the quaternary structure.

Cf. A134144 (M_3(3) array).

Sequence in context: A134151 A119304 A114150 this_sequence A035469 A073323 A077097

Adjacent sequences: A134146 A134147 A134148 this_sequence A134150 A134151 A134152

nonn,easy,tabf

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Nov 13 2007