OFFSET
1,2
COMMENTS
The representation of the partitions (for fixed n) is as (weakly) increasing lists of parts, the order between individual partitions (for the same n) is (list-)reversed lexicographic; see examples. [Joerg Arndt, Sep 03 2013]
Also compositions in the triangle of A066099 that are in nondecreasing order.
The equivalent sequence for compositions (ordered partitions) is A066099.
Row n has length A006128(n).
Row sums give A066186.
LINKS
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order
EXAMPLE
Illustration of initial terms:
---------------------------------
. Ordered
n j Diagram partition
---------------------------------
. _
1 1 |_| 1;
. _ _
2 1 | _| 2,
2 2 |_|_| 1, 1;
. _ _ _
3 1 | _ _| 3,
3 2 | | _| 1, 2,
3 3 |_|_|_| 1, 1, 1;
. _ _ _ _
4 1 | _ _| 4,
4 2 | _|_ _| 2, 2,
4 3 | | _ _| 1, 3,
4 4 | | | _| 1, 1, 2,
4 5 |_|_|_|_| 1, 1, 1, 1;
.
Triangle begins:
[1];
[2],[1,1];
[3],[1,2],[1,1,1];
[4],[2,2],[1,3],[1,1,2],[1,1,1,1];
[5],[2,3],[1,4],[1,2,2],[1,1,3],[1,1,1,2],[1,1,1,1,1];
[6],[3,3],[2,4],[2,2,2],[1,5],[1,2,3],[1,1,4],[1,1,2,2],[1,1,1,3],[1,1,1,1,2],[1,1,1,1,1,1];
[7],[3,4],[2,5],[2,2,3],[1,6],[1,3,3],[1,2,4],[1,2,2,2],[1,1,5],[1,1,2,3],[1,1,1,4],[1,1,1,2,2],[1,1,1,1,3],[1,1,1,1,1,2],[1,1,1,1,1,1,1];
...
MATHEMATICA
revlexsort[f_, c_]:=OrderedQ[PadRight[{c, f}]];
Join@@Table[Sort[Reverse/@IntegerPartitions[n], revlexsort], {n, 0, 8}] (* Gus Wiseman, May 23 2020 *)
CROSSREFS
Row lengths are A000041.
Partition sums are A036042.
Partition minima are A182715.
Partition lengths are A333486.
The lexicographic version (sum/lex) is A026791.
Compositions under the same order (sum/revlex) are A066099.
The colexicographic version (sum/colex) is A080576.
The version for non-reversed partitions is A080577.
The length-sensitive version (sum/length/revlex) is A334302.
The Heinz numbers of these partitions are A334436.
Partitions in colexicographic order (sum/colex) are A211992.
Partitions in lexicographic order (sum/lex) are A193073.
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Aug 30 2013
STATUS
approved