

A080577


Triangle in which nth row lists all partitions of n, in graded reverse lexicographic ordering.


69



1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 4, 3, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 5, 4, 1, 3, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 1, 4, 2, 4, 1, 1, 3, 3, 3, 2, 1, 3, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 6, 1, 5, 2, 5, 1, 1, 4, 3, 4, 2, 1, 4, 1, 1, 1, 3, 3, 1, 3, 2
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OFFSET

1,2


COMMENTS

This is the "Mathematica" ordering of the partitions, referenced in numerous other sequences. The partitions of each integer are in reverse order of the conjugates of the partitions in Abramowitz and Stegun order (A036036). They are in the reverse of the order of the partitions in Maple order (A080576).  Franklin T. AdamsWatters, Oct 18 2006
The graded reverse lexicographic ordering of the partitions is often referred to as the "Canonical" ordering of the partitions.  Daniel Forgues, Jan 21 2011
Also the "MAGMA" ordering of the partitions.  Jason Kimberley, Oct 28 2011
Also an intuitive ordering described but not formalized in [Hardy and Wright] the first four editions of which precede [Abramowitz and Stegun].  L. Edson Jeffery, Aug 03 2013
Also the "Sage" ordering of the partitions.  Peter Luschny, Aug 12 2013


REFERENCES

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Oxford, Fifth edition, 1979, p. 273.


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, p. 831.
OEIS Wiki, Orderings of partitions (a comparison).
Sergei Viznyuk, C Program


EXAMPLE

First five rows are:
{{1}}
{{2}, {1, 1}}
{{3}, {2, 1}, {1, 1, 1}}
{{4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}}
{{5}, {4, 1}, {3, 2}, {3, 1, 1}, {2, 2, 1}, {2, 1, 1, 1}, {1, 1, 1, 1, 1}}
Up to the fifth row, this is exactly the same as the colexicographic ordering A036037. The first row which differs is the sixth one, which reads ((6), (5,1), (4,2), (4,1,1), (3,3), (3,2,1), (3,1,1,1), (2,2,2), (2,2,1,1), (2,1,1,1,1), (1,1,1,1,1,1)).  M. F. Hasler, Jan 23 2020


MAPLE

b:= (n, i)> `if`(n=0 or i=1, [[1$n]], [map(x>
[i, x[]], b(ni, min(ni, i)))[], b(n, i1)[]]):
T:= n> map(x> x[], b(n$2))[]:
seq(T(n), n=1..8); # Alois P. Heinz, Jan 29 2020


MATHEMATICA

<<DiscreteMath`Combinatorica`; Partition[6]
(* Or, from version 6 on : *) Table[ IntegerPartitions[n], {n, 1, 7}] // Flatten (* JeanFrançois Alcover, Dec 10 2012 *)


PROG

(MAGMA) &cat[&cat Partitions(n):n in[1..7]]; // Jason Kimberley, Oct 28 2011
(Sage)
L = []
for n in range(8): L += list(Partitions(n))
flatten(L) # Peter Luschny, Aug 12 2013
(PARI) A080577_row(n)={vecsort(apply(t>Vecrev(t), partitions(n)), , 4)} \\ M. F. Hasler, Jan 21 2020


CROSSREFS

See A080576 Maple (graded reflected lexicographic) ordering.
See A036036 for the Hindenburg (graded reflected colexicographic) ordering (listed in the Abramowitz and Stegun Handbook).
See A036037 for graded colexicographic ordering.
See A228100 for the FennerLoizou (binary tree) ordering.
Differs from A036037 at a(48).
See A322761 for a compressed version.
Sequence in context: A036037 A181317 A330370 * A302246 A209655 A209918
Adjacent sequences: A080574 A080575 A080576 * A080578 A080579 A080580


KEYWORD

nonn,tabf


AUTHOR

N. J. A. Sloane, Mar 23 2003


STATUS

approved



