A080576 Triangle in which n-th row lists all partitions of n, in graded reflected lexicographic order.

1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 2, 3, 1, 4, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 3, 1, 2, 3, 3, 3, 1, 1, 4, 2, 4, 1, 5, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2

Offset: 1

Author

N. J. A. Sloane, Mar 23 2003

Keywords

• nonn
• tabf

Comments

The graded reflected lexicographic ordering of the partitions is used by Maple. - Daniel Forgues, Jan 19 2011

Each partition here is the conjugate of the corresponding partition in Abramowitz and Stegun order (A036036). The partitions are in the reverse of the order of the partitions in Mathematica order (A080577). - Franklin T. Adams-Watters, Oct 18 2006

Reversing all partitions gives A193073 (the non-reflected version). The version for reversed (weakly increasing) partitions is A211992. Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036. - Gus Wiseman, May 20 2020

Also reversed integer partitions in colexicographic order, cf. A228531. - Gus Wiseman, May 31 2020

Examples

			First five rows are:
[[1]]
[[1, 1], [2]]
[[1, 1, 1], [1, 2], [3]]
[[1, 1, 1, 1], [1, 1, 2], [2, 2], [1, 3], [4]]
[[1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 2, 2], [1, 1, 3], [2, 3], [1, 4], [5]]
From _Gus Wiseman_, May 20 2020: (Start)
The sequence of all reversed partitions begins:
  ()       (122)     (15)       (25)
  (1)      (113)     (6)        (16)
  (11)     (23)      (1111111)  (7)
  (2)      (14)      (111112)   (11111111)
  (111)    (5)       (11122)    (1111112)
  (12)     (111111)  (1222)     (111122)
  (3)      (11112)   (11113)    (11222)
  (1111)   (1122)    (1123)     (2222)
  (112)    (222)     (223)      (111113)
  (22)     (1113)    (133)      (11123)
  (13)     (123)     (1114)     (1223)
  (4)      (33)      (124)      (1133)
  (11111)  (114)     (34)       (233)
  (1112)   (24)      (115)      (11114)
(End)
		

Links

• Alois P. Heinz, Rows n = 1..20, 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)
• A. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions
• Wikiversity, Lexicographic and colexicographic order

Crossrefs

See A080577 for the Mathematica (graded reverse lexicographic) ordering.

See A036036 for the Hindenburg (graded reflected colexicographic) ordering (listed in the Abramowitz and Stegun Handbook).

See A036037 for the graded colexicographic ordering.

See A193073 for the graded lexicographic ordering. - M. F. Hasler, Jul 16 2011

See A228100 for the Fenner-Loizou (binary tree) ordering.

Row n has A000041(n) partitions.

Taking colexicographic instead of lexicographic gives A026791.

Lengths of these partitions appear to be A049085.

Reversing all partitions gives A193073 (the non-reflected version).

The version for reversed (weakly increasing) partitions is A211992.

The generalization to compositions is A228525.

The Heinz numbers of these partitions are A334434.

Cf. A026791, A036037, A112798, A129129, A185974, A228351, A228531, A334301, A334302, A334433, A334437.

Programs

Maple

with(combinat); partition(6);

Mathematica

row[n_] := Flatten[Reverse /@ Reverse[SplitBy[Reverse /@ IntegerPartitions[n], Length]], 1]; Array[row, 7] // Flatten (* Jean-François Alcover, Dec 05 2016 *)
lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
Reverse/@Join@@Table[Sort[IntegerPartitions[n],lexsort],{n,0,8}] (* Gus Wiseman, May 20 2020 *)

Extensions

Edited by Daniel Forgues, Jan 21 2011