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%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
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