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A130561
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Numbers associated to partitions, used for combinatoric interpretation of Lah triangle numbers A105278.
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13
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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
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OFFSET
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1,2
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COMMENTS
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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] (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.
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.
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LINKS
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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].
Tom Copeland, Lagrange a la Lah
W. Lang, First 10 rows and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.
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FORMULA
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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.
Contribution 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)
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EXAMPLE
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[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.
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CROSSREFS
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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 * A157400 A091599 A048999
Adjacent sequences: A130558 A130559 A130560 * A130562 A130563 A130564
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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Wolfdieter Lang Jul 13 2007
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STATUS
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approved
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