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A130561 Numbers associated to partitions, used for combinatoric interpretation of Lah triangle numbers A105278. 13
1, 2, 1, 6, 6, 1, 24, 24, 12, 12, 1, 120, 120, 120, 60, 60, 20, 1, 720, 720, 720, 360, 360, 720, 120, 120, 180, 30, 1, 5040, 5040, 5040, 5040, 2520, 5040, 2520, 2520, 840, 2520, 840, 210, 420, 42, 1, 40320, 40320, 40320, 40320, 20160, 20160, 40320, 40320, 20160 (list; graph; refs; listen; history; text; internal format)



The order of this array is according to the Abramowitz-Stegun (A-St) ordering of partitions (see A036036).

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] (5th 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). - Wolfdieter 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. - Wolfdieter Lang, Oct 19 2007

A relation between partition polynomials formed from these "refined Lah numbers" and Lagrange inversion for an o.g.f. is presented in the link "Lagrange a la Lah" along with an e.g.f. and an umbral binary operator tree representation. - Tom Copeland, Apr 12 2011


Table of n, a(n) for n=1..53.

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].

T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms

T. Copeland, Lagrange a la Lah

Wolfdieter  Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Wolfdieter 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 partitions of n. Exponents 0 can be omitted due to 0!=1.

From Tom Copeland, Sep 18 2011: (Start)

Raising and lowering operators are given for the partition polynomials formed from A130561 in the Copeland link in "Lagrange a la Lah Part I" on pg. 22-23.

An e.g.f. for the partition polynomials is on page 3:

  exp[t*:c.*x/(1-c.*x):] = exp[t*(c_1*x + c_2*x^2 + c_3*x^3 + ...)] where:(...): denotes umbral evaluation of the enclosed expression and c. is an umbral coefficient. (End)


[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 6th 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). - Wolfdieter Lang, Oct 12 2007

Sequence in context: A110183 A110098 A244888 * A157400 A091599 A048999

Adjacent sequences:  A130558 A130559 A130560 * A130562 A130563 A130564




Wolfdieter Lang, Jul 13 2007



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Last modified September 26 01:05 EDT 2016. Contains 276541 sequences.