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Search: id:A080575
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| A080575 |
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Triangle of multinomial coefficients, read by rows. |
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+0
6
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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
(list; graph; listen)
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OFFSET
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1,5
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COMMENT
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T[n,m]=count of set partitions of n with block lengths given by the m-th partition of n.
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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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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].
Wikipedia, Cumulant.
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EXAMPLE
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1; 1,1; 1,3,1; 1,4,3,6,1; ...
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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CROSSREFS
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See A036040 for another version. Cf. A036036-A036039.
Row sums are A000110.
Sequence in context: A049999 A126015 A036040 this_sequence A077228 A049687 A132735
Adjacent sequences: A080572 A080573 A080574 this_sequence 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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