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A080577 Triangle in which n-th row lists all partitions of n, in graded reverse lexicographic ordering. 50
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 (list; graph; refs; listen; history; text; internal format)
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. Adams-Watters, 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. 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, 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}}

MATHEMATICA

<<DiscreteMath`Combinatorica`; Partition[6]

(* Or, from version 6 on : *) Table[ IntegerPartitions[n], {n, 1, 7}] // Flatten (* Jean-Fran├žois 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

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 Fenner-Loizou (binary tree) ordering.

Differs from A036037 at a(48).

Sequence in context: A239512 A036037 A181317 * A209655 A209918 A030312

Adjacent sequences:  A080574 A080575 A080576 * A080578 A080579 A080580

KEYWORD

nonn,tabf

AUTHOR

N. J. A. Sloane, Mar 23 2003

STATUS

approved

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Last modified November 18 19:06 EST 2017. Contains 294894 sequences.