

A036036


Triangle read by rows in which row n lists all the parts of all the partitions of n, in graded reflected colexicographic ordering.


66



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
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OFFSET

1,2


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. AdamsWatters, 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. 775 with the listing of the partitions for n=10. This is also mentioned in the P. Luschny link.  Wolfdieter Lang, Apr 04 2011
More explicitly, the "Abramowitz and Stegun" order means that the partitions of a given number are listed by increasing number of parts, then by decreasing lexicographical order of the parts read from right to left, where the parts are listed in increasing order. Otherwise said, list the parts in decreasing order, then order the partitions by number of parts and then in decreasing lexicographical order, then reverse the order of the parts in each partition. For example, in row 6, one has ...; 3,3; 1,1,4; 1,2,3; 2,2,2; 1,1,1,3; ...  M. F. Hasler, Jul 12 2015


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, AddisonWesley, 2005.


LINKS

Franklin T. AdamsWatters, 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).


EXAMPLE

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;
...


MATHEMATICA

row[n_] := Flatten[SplitBy[Sort[Reverse /@ IntegerPartitions[n]], Length], 1]; Array[row, 7] // Flatten (* JeanFrançois Alcover, May 19 2011, updated Dec 05 2016 *)


PROG

(PARI) T036036(n, k)=k&&return(T036036(n)[k]); concat(partitions(n))
\\ If 2nd arg "k" is not given, return the nth row as a vector. Assumes PARI version >= 2.7.1. See A193073 for "hand made" code.
concat(vector(8, n, T036036(n))) \\ to get the "flattened" sequence
\\ M. F. Hasler, Jul 12 2015


CROSSREFS

Cf. A036037A036040.
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.
See A193073 for the graded lexicographic ordering.
See A228100 for the FennerLoizou (binary tree) ordering.
Sequence in context: A164659 A057898 A094293 * A228531 A244316 A076259
Adjacent sequences: A036033 A036034 A036035 * A036037 A036038 A036039


KEYWORD

nonn,easy,nice,tabf,look


AUTHOR

N. J. A. Sloane


EXTENSIONS

Comment by Daniel Forgues, Jan 19 2011
Edited by Daniel Forgues, Jan 21 2011
Mathematica program updated by Harvey P. Dale, Dec 30 2011
Edited by M. F. Hasler, Jul 12 2015


STATUS

approved



