1,2

The order of this array is according to the Abramowitz-Stegun (A-St) ordering of partitions (for the reference see A117506).

The row lengths sequence is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].

These numbers are similar to M_0, M_1, M_2, M_3, M_4 given in A111786, A036038, A036039, A036040, A117506, resp.).

Combinatorial interpretation: a(n,k) counts the sets of lists (ordered subsets) obtained from partitioning the set {1,2,...,n}, with the lengths of the lists given by the k-th partition of n in A-St order. E.g. a(5,5) is computed from the number of sets of lists of lengths [1^1,2^2] (5-th partition of 5 in A-St order). Hence a(5,5)=binomial(5,2)*binomial(3,2)= 5!/(1!*2!)=60 from partitioning the numbers 1,2,...,5 into sets of lists of the type {[.],[..],[..]}.

This array, called M_3(2), is the k=2 member of a family of partition arrays generalizing A036040 which appears as M_3 = M_3(k=1). S2(2)= A105278 (unsigned Lah number triangle) is related to M_3(2) in the same way as S2(1), the Stirling2 number triangle, is related to M_3(1). W. Lang, Oct 19 2007.

Another combinatorial interpretation: a(n,k) enumerates unordered forests of increasing binary trees which are described by the k-th partition of n in the Abramowitz-Stegun order. W. Lang, Oct 19 2007.

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(e(n,k,j)!,j=1..n) with the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitons of n. Exponents 0 can be omitted due to 0!=1.

[1]; [2,1]; [6,6,1]; [24,24,12,12,1]; [120,120,120,60,60,20,1];...

a(5,6)= 20 = 5!/(3!*1!) because the 6-th partition of 5 in A-St order is [1^3,2^1].

a(5,5)=60 enumerates the unordered [1^1,2^2]-forest with 5 vertices (including the three roots) composed of three such increasing binary trees: 5*((binomial(4,2)*2)*(1*2))/2!=5*12=60.

Cf. A105278 (unsigned Lah triangle |L(n, m)|)obtained by summing the numbers for given part number m.

Cf. A000262 (row sums), identical with row sums of unsigned Lah triangle A105278.

A134133(n, k) = A130561(n, k)/A036040(n, k) (division by the M_3 numbers). W. Lang, Oct 12 2007.

Sequence in context: A060538 A110183 A110098 this_sequence A091599 A066667 A105278

Adjacent sequences: A130558 A130559 A130560 this_sequence A130562 A130563 A130564

nonn,tabf,easy

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