A026792 List of juxtaposed reverse-lexicographically ordered partitions of the positive integers.

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

Offset

1,2

Comments

The representation of the partitions (for fixed n) is as (weakly) decreasing lists of parts, the order between individual partitions (for the same n) is (list-)reversed lexicographic; see examples. [Joerg Arndt, Sep 03 2013]

Written as a triangle; row n has length A006128(n); row sums give A066186. Also written as an irregular tetrahedron in which T(n,j,k) is the k-th largest part of the j-th partition of n; the sum of column k in the slice n is A181187(n,k); right border of the slices gives A182715. - Omar E. Pol, Mar 25 2012

The equivalent sequence for compositions (ordered partitions) is A228351). - Omar E. Pol, Sep 03 2013

This is the reverse-colexicographic order of integer partitions, or the reflected reverse-lexicographic order of reversed integer partitions. It is not reverse-lexicographic order (A080577), wherein we would have (3,1) before (2,2). - Gus Wiseman, May 12 2020

Links

Robert Price, Table of n, a(n) for n = 1..3615, 15 rows.

OEIS Wiki, Orderings of partitions

Wikiversity, Lexicographic and colexicographic order

Example

E.g. the partitions of 3 (3,2+1,1+1+1) appear as the string 3,2,1,1,1,1.
So the list begins:
1
2, 1, 1,
3, 2, 1, 1, 1, 1,
4, 2, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1,
5, 3, 2, 4, 1, 2, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1,
...
From Omar E. Pol, Sep 03 2013: (Start)
Illustration of initial terms:
---------------------------------
n  j     Diagram     Partition
---------------------------------
.         _
1  1     |_|         1;
.         _ _
2  1     |_  |       2,
2  2     |_|_|       1, 1;
.         _ _ _
3  1     |_ _  |     3,
3  2     |_  | |     2, 1,
3  3     |_|_|_|     1, 1, 1;
.         _ _ _ _
4  1     |_ _    |   4,
4  2     |_ _|_  |   2, 2,
4  3     |_ _  | |   3, 1,
4  4     |_  | | |   2, 1, 1,
4  5     |_|_|_|_|   1, 1, 1, 1;
...
(End)
From Gus Wiseman, May 12 2020: (Start)
This sequence can also be interpreted as the following triangle, whose n-th row is itself a finite triangle with A000041(n) rows. Showing these partitions as their Heinz numbers gives A334436.
                             0
                            (1)
                          (2)(11)
                        (3)(21)(111)
                   (4)(22)(31)(211)(1111)
             (5)(32)(41)(221)(311)(2111)(11111)
  (6)(33)(42)(222)(51)(321)(411)(2211)(3111)(21111)(111111)
(End)

Mathematica

revcolex[f_, c_]:=OrderedQ[PadRight[{Reverse[c], Reverse[f]}]];
Join@@Table[Sort[IntegerPartitions[n], revcolex], {n, 0, 8}] (* reverse-colexicographic order, Gus Wiseman, May 10 2020 *)
- or -
revlex[f_, c_]:=OrderedQ[PadRight[{c, f}]];
Reverse/@Join@@Table[Sort[Reverse/@IntegerPartitions[n], revlex], {n, 0, 8}] (* reflected reverse-lexicographic order, Gus Wiseman, May 12 2020 *)

Crossrefs

Cf. A026791, A228531.

The reflected version for reversed partitions is A080577.

The partition minima appear to be A182715.

The graded reversed version is A211992.

The version for compositions is A228351.

The Heinz numbers of these partitions are A334436.

Cf. A000041, A036036, A036037, A193073, A296150, A331581, A334301, A334435, A334437, A334439.

Sequence in context: A211028 A239001 A277648 * A334301 A139100 A237982

Adjacent sequences: A026789 A026790 A026791 * A026793 A026794 A026795

Keyword

nonn,tabf

Author

Clark Kimberling

Extensions

Terms 81st, 83rd and 84th corrected by Omar E. Pol, Aug 16 2009

Status

approved