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A080575
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Triangle of multinomial coefficients, read by rows (version 2).
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13
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1, 1, 1, 1, 3, 1, 1, 4, 3, 6, 1, 1, 5, 10, 10, 15, 10, 1, 1, 6, 15, 15, 10, 60, 20, 15, 45, 15, 1, 1, 7, 21, 21, 35, 105, 35, 70, 105, 210, 35, 105, 105, 21, 1, 1, 8, 28, 28, 56, 168, 56, 35, 280, 210, 420, 70, 280, 280, 840, 560, 56, 105, 420, 210, 28, 1, 1, 9, 36, 36, 84, 252, 84, 126, 504, 378, 756, 126, 315, 1260, 1260, 1890, 1260, 126, 280, 2520, 840, 1260, 3780, 1260, 84, 945, 1260, 378, 36, 1, 1, 10, 45, 45, 120, 360, 120, 210, 840, 630, 1260, 210
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OFFSET
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1,5
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COMMENTS
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This is different from A036040 and A178867.
T[n,m] = count of set partitions of n with block lengths given by the m-th partition of n in the canonical ordering.
Contribution from Tilman Neumann, Oct 05 2008: (Start)
These are also the coefficients occuring in complete Bell polynomials, Faa di Bruno's formula (in it's simplest form) and computation of moments from cumulants.
Though the Bell polynomials seem quite unwieldy, they can be computed easily as the determinant of an n-dimensional square matrix. (see e.g. [Coffey] and program below)
The complete Bell polynomial of the first n primes gives A007446. (End)
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REFERENCES
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See A036040 for the column labeled "M_3" in Abramowitz and Stegun, Handbook, p. 831.
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LINKS
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Alois P. Heinz, Rows n = 1..26, flattened
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].
Mark W. Coffey, A Set of Identities for a Class of Alternating Binomial Sums Arising in Computing Applications
Wikipedia, Cumulant.
Wikipedia, Bell polynomials
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EXAMPLE
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For n=4 the 5 integer partitions in canonical ordering with corresponding set partitions and counts are:
[4] -> #{1234} = 1
[3,1] -> #{123/4, 124/3, 134/2, 1/234} = 4
[2,2] -> #{12/34, 13/24, 14/23} = 3
[2,1,1] -> #{12/3/4, 13/2/4, 1/23/4, 14/2/3, 1/24/3, 1/2/34} = 6
[1,1,1,1] -> #{1/2/3/4} = 1
Thus row 4 is [1, 4, 3, 6, 1].
Triangle begins:
1;
1, 1;
1, 3, 1;
1, 4, 3, 6, 1;
1, 5, 10, 10, 15, 10, 1;
1, 6, 15, 15, 10, 60, 20, 15, 45, 15, 1;
1, 7, 21, 21, 35, 105, 35, 70, 105, 210, 35, 105, 105, 21, 1;
...
Row 4 represents 1*k(4)+4*k(3)*k(1)+3*k(2)^2+6*k(2)*k(1)^2+1*k(1)^4 and T(4,4)=6 since there are six ways of partitioning four labeled items into one part with two items and two parts each with one item.
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MATHEMATICA
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<<DiscreteMath`Combinatorica`; runs[li:{__Integer}] := ((Length/@ Split[ # ]))&[Sort@ li]; Table[Apply[Multinomial, Partitions[w], {1}]/Apply[Times, (runs/@ Partitions[w])!, {1}], {w, 6}]
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PROG
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Contribution from Tilman Neumann, Oct 05 2008: (Start)
(MuPAD)
completeBellMatrix := proc(x, n)
// x - vector x[1]...x[m], m>=n
local i, j, M;
begin
M:=matrix(n, n): // zero-initialized
for i from 1 to n-1 do
M[i, i+1]:=-1:
end_for:
for i from 1 to n do
for j from 1 to i do
M[i, j] := binomial(i-1, j-1)*x[i-j+1]:
end_for:
end_for:
return (M):
end_proc:
completeBellPoly := proc(x, n)
begin
return (linalg::det(completeBellMatrix(x, n))):
end_proc:
for i from 1 to 10 do print(i, completeBellPoly(x, i)): end_for:
(End)
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CROSSREFS
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See A036040 for another version. Cf. A036036-A036039.
Row sums are A000110.
Row lengths are A000041.
Cf. A007446 [From Tilman Neumann, Oct 05 2008]
Cf. A178866 and A178867 (version 3). [From Johannes W. Meijer, June 21, 2010]
Maximum value in row n gives A102356(n).
Sequence in context: A126015 A144336 A036040 * A205117 A077228 A049687
Adjacent sequences: A080572 A080573 A080574 * A080576 A080577 A080578
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KEYWORD
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nonn,easy,nice,tabf
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AUTHOR
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Wouter Meeussen, Mar 23 2003
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STATUS
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approved
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