1,2

For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.

Partition number array M_3(6), the k=6 member in the family of a generalization of the multinomial number arrays 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(6,n,m):=A049385(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 6-ary 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(6,j,1)/j!)^e(n,k,j)/e(n,k,j)!,j=1..n) with S2(6,n,1)=A049385(n,1) = A008548(n) = (5*n-4)(!^5) (quintupel- or 5-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];[6,1];[66,18,1];[1056,264,108,36,1];[22176,5280,3960,660,540,60,1];...

There are a(4,3)=108=3*6^2 unordered 2-forest with 4 vertices, composed of two 6-ary increasing 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 six versions from the 6-ary structure.

Cf. A049412 (row sums, also of triangle A049385).

Cf. A134273 (M_3(5) partition array).

Adjacent sequences: A134275 A134276 A134277 this_sequence A134279 A134280 A134281

Sequence in context: A056218 A134279 A134280 this_sequence A049385 A009384 A051151

nonn,easy,tabf

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