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A036036
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Triangle read by rows in which row n lists all the parts of all the partitions of n, in graded reflected colexicographic ordering.
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49
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1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 3, 2, 2, 1, 1, 2, 1, 1, 1, 1, 5, 1, 4, 2, 3, 1, 1, 3, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 6, 1, 5, 2, 4, 3, 3, 1, 1, 4, 1, 2, 3, 2, 2, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 7, 1, 6, 2, 5, 3, 4, 1, 1, 5, 1, 2, 4, 1, 3, 3, 2, 2, 3, 1, 1, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| This is the "Abramowitz and Stegun" ordering of the partitions, referenced in numerous other sequences. The partitions are in reverse order of the conjugates of the partitions in Mathematica order (A080577). Each partition is the conjugate of the corresponding partition in Maple order (A080576). - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Oct 18 2006
The "Abramowitz and Stegun" ordering of the partitions is the graded reflected colexicographic ordering of the partitions. - Daniel Forgues, Jan 19 2011
The "Abramowitz and Stegun" ordering of partitions has been traced back to C. F. Hindenburg, 1779, in the Knuth reference, p. 38. See the Hindenburg link, pp.77-5 with the listing of the partitions for n=10. This is also mentioned in the P. Luschny link. - W. Lang, Apr 04 2011.
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REFERENCES
| Abramowitz and Stegun, Handbook, p. 831, column labeled "pi".
D. Knuth, The Art of Computer Programming, Vol. 4, fascicle 3, 7.2.1.4, Addison-Wesley, 2005.
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LINKS
| Franklin T. Adams-Watters, First 20 rows, 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]. (uses Flash)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions.
C. F. Hindenburg, Infinitinomii Dignitatum Exponentis Indeterminati, Goettingen 1779.
P. Luschny, Counting with partitions.
OEIS Wiki, Orderings of partitions (a comparison).
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EXAMPLE
| 1 / 2; 1,1 / 3; 1,2; 1,1,1 / 4; 1,3; 2,2; 1,1,2; 1,1,1,1 / ...
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MATHEMATICA
| f[n_] := Sort /@ Split[SortBy[Reverse /@ IntegerPartitions[n], Length], Length[#1] == Length[#2] &]; Flatten[f /@ Range[7]][[1 ;; 99]]
(* From Jean-François Alcover, May 19 2011 *)
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CROSSREFS
| Cf. A036037-A036040.
See A036037 for the graded colexicographic ordering.
See A080576 for the Maple (graded reflected lexicographic) ordering.
See A080577 for the Mathematica (graded reverse lexicographic) ordering.
Sequence in context: A164659 A057898 A094293 * A076259 A107359 A112377
Adjacent sequences: A036033 A036034 A036035 * A036037 A036038 A036039
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KEYWORD
| nonn,easy,nice,tabf,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Comment by Daniel Forgues (kephalopod(AT)gmail.com), Jan 19 2011
Edited by Daniel Forgues (kephalopod(AT)gmail.com), Jan 21 2011
Updated Mathematica program [Harvey P. Dale, Dec 30 2011]
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