A228531 Triangle read by rows in which row n lists the partitions of n in reverse lexicographic order.

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

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

Table of n, a(n) for n=1..87.

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.

Cf. A026792, A036036, A049085, A103921, A112798, A115623, A129129, A228351, A331581, A334435, A334439, A334442.

Sequence in context: A036036 A344091 A334302 * A360056 A244316 A076259

Adjacent sequences: A228528 A228529 A228530 * A228532 A228533 A228534

Keyword

nonn,tabf

Author

Omar E. Pol, Aug 30 2013

Status

approved