login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A080577 Triangle in which n-th row lists all partitions of n, in graded reverse lexicographic ordering. 94
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

While this is the order used for the constructive function "IntegerPartitions", it is different from Mathematica's canonical ordering of finite expressions, the latter giving A036036 if parts of partitions are read in reversed (weakly increasing) order, or A334301 if in the usual (weakly decreasing) order. - Gus Wiseman, May 08 2020

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

Wikiversity, Lexicographic and colexicographic order

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

From Gus Wiseman, May 08 2020: (Start)

The sequence of all partitions begins:

  ()         (3,2)        (2,1,1,1,1)    (2,2,1,1,1)

  (1)        (3,1,1)      (1,1,1,1,1,1)  (2,1,1,1,1,1)

  (2)        (2,2,1)      (7)            (1,1,1,1,1,1,1)

  (1,1)      (2,1,1,1)    (6,1)          (8)

  (3)        (1,1,1,1,1)  (5,2)          (7,1)

  (2,1)      (6)          (5,1,1)        (6,2)

  (1,1,1)    (5,1)        (4,3)          (6,1,1)

  (4)        (4,2)        (4,2,1)        (5,3)

  (3,1)      (4,1,1)      (4,1,1,1)      (5,2,1)

  (2,2)      (3,3)        (3,3,1)        (5,1,1,1)

  (2,1,1)    (3,2,1)      (3,2,2)        (4,4)

  (1,1,1,1)  (3,1,1,1)    (3,2,1,1)      (4,3,1)

  (5)        (2,2,2)      (3,1,1,1,1)    (4,2,2)

  (4,1)      (2,2,1,1)    (2,2,2,1)      (4,2,1,1)

The triangle with partitions shown as Heinz numbers (A129129) begins:

   1

   2

   3   4

   5   6   8

   7  10   9  12  16

  11  14  15  20  18  24  32

  13  22  21  28  25  30  40  27  36  48  64

  17  26  33  44  35  42  56  50  45  60  80  54  72  96 128

(End)

MAPLE

b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->

    [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):

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 (* Jean-François Alcover, Dec 10 2012 *)

revlexsort[f_, c_]:=OrderedQ[PadRight[{c, f}]];

Join@@Table[Sort[IntegerPartitions[n], revlexsort], {n, 0, 8}] (* Gus Wiseman, May 08 2020 *)

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 Fenner-Loizou (binary tree) ordering.

Differs from A036037 at a(48).

See A322761 for a compressed version.

Lexicographically ordered reversed partitions are A026791.

Reverse-colexicographically ordered partitions are A026792.

Compositions under this ordering are A066099.

Distinct parts of these partitions are counted by A115623.

Taking Heinz numbers gives A129129.

Lexicographically ordered partitions are A193073.

Colexicographically ordered partitions are A211992.

Reading partitions in reverse (weakly increasing) order gives A228531.

Lengths of these partitions are A238966.

Sorting partitions by Heinz number gives A296150.

The maxima of these partitions are A331581.

The length-sensitive version is A334439.

Cf. A000041, A048793, A063008, A185974, A334301, A334434, A334436, A334438.

Sequence in context: A181317 A330370 A330371 * A302246 A209655 A209918

Adjacent sequences:  A080574 A080575 A080576 * A080578 A080579 A080580

KEYWORD

nonn,tabf

AUTHOR

N. J. A. Sloane, Mar 23 2003

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 2 19:12 EST 2020. Contains 338891 sequences. (Running on oeis4.)