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Triangle read by rows in which row n lists the partitions of n in reverse lexicographic order.
34

%I #37 Sep 22 2023 07:54:45

%S 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,

%T 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,

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

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

%C 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]

%C Also compositions in the triangle of A066099 that are in nondecreasing order.

%C The equivalent sequence for compositions (ordered partitions) is A066099.

%C Row n has length A006128(n).

%C Row sums give A066186.

%H OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>

%H Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>

%e Illustration of initial terms:

%e ---------------------------------

%e . Ordered

%e n j Diagram partition

%e ---------------------------------

%e . _

%e 1 1 |_| 1;

%e . _ _

%e 2 1 | _| 2,

%e 2 2 |_|_| 1, 1;

%e . _ _ _

%e 3 1 | _ _| 3,

%e 3 2 | | _| 1, 2,

%e 3 3 |_|_|_| 1, 1, 1;

%e . _ _ _ _

%e 4 1 | _ _| 4,

%e 4 2 | _|_ _| 2, 2,

%e 4 3 | | _ _| 1, 3,

%e 4 4 | | | _| 1, 1, 2,

%e 4 5 |_|_|_|_| 1, 1, 1, 1;

%e .

%e Triangle begins:

%e [1];

%e [2],[1,1];

%e [3],[1,2],[1,1,1];

%e [4],[2,2],[1,3],[1,1,2],[1,1,1,1];

%e [5],[2,3],[1,4],[1,2,2],[1,1,3],[1,1,1,2],[1,1,1,1,1];

%e [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];

%e [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];

%e ...

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

%t Join@@Table[Sort[Reverse/@IntegerPartitions[n],revlexsort],{n,0,8}] (* _Gus Wiseman_, May 23 2020 *)

%Y Row lengths are A000041.

%Y Partition sums are A036042.

%Y Partition minima are A182715.

%Y Partition lengths are A333486.

%Y The lexicographic version (sum/lex) is A026791.

%Y Compositions under the same order (sum/revlex) are A066099.

%Y The colexicographic version (sum/colex) is A080576.

%Y The version for non-reversed partitions is A080577.

%Y The length-sensitive version (sum/length/revlex) is A334302.

%Y The Heinz numbers of these partitions are A334436.

%Y Partitions in colexicographic order (sum/colex) are A211992.

%Y Partitions in lexicographic order (sum/lex) are A193073.

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

%K nonn,tabf

%O 1,2

%A _Omar E. Pol_, Aug 30 2013