1,2

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

Partition number array M_3(3), the k=3 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(3,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 ternary trees related to the k-th partition of n in the A-St order. The 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, 1972 [alternative scanned copy].

W. Lang, First 10 rows and more.

a(n,k)= n!*product((S2(3,j,1)/j!)^e(n,k,j)/e(n,k,j)!,j=1..n) with S2(3,n,1)=A035342(n,1) = A001147(n) = (2*n-1)!! 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]; [3,1]; [15,9,1]; [105,60,27,18,1]; [945,525,450,150,135,30,1];...

a(4,3)=27 from the partition (2^2) of 4: 4!*((3/2!)^2)/2! = 27.

There are a(4,3)=27= 3*3^2 unordered 2-forest with 4 vertices, composed of two increasing ternary 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 three versions from the ternary structure.

Cf. A049118 (row sums, identical with those of triangle A035342).

Sequence in context: A113389 A038553 A135896 this_sequence A035342 A039815 A147453

Adjacent sequences: A134141 A134142 A134143 this_sequence A134145 A134146 A134147

nonn,easy,tabf

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