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

%I A228531 #37 Sep 22 2023 07:54:45
%S A228531 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 A228531 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 A228531 1,1,2,1,1,1,1,1,1,7,3,4,2,5,2,2,3,1,6
%N A228531 Triangle read by rows in which row n lists the partitions of n in reverse lexicographic order.
%C A228531 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 A228531 Also compositions in the triangle of A066099 that are in nondecreasing order.
%C A228531 The equivalent sequence for compositions (ordered partitions) is A066099.
%C A228531 Row n has length A006128(n).
%C A228531 Row sums give A066186.
%H A228531 OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>
%H A228531 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%e A228531 Illustration of initial terms:
%e A228531 ---------------------------------
%e A228531 .                    Ordered
%e A228531 n  j     Diagram     partition
%e A228531 ---------------------------------
%e A228531 .              _
%e A228531 1  1          |_|    1;
%e A228531 .            _ _
%e A228531 2  1        |  _|    2,
%e A228531 2  2        |_|_|    1, 1;
%e A228531 .          _ _ _
%e A228531 3  1      |  _ _|    3,
%e A228531 3  2      | |  _|    1, 2,
%e A228531 3  3      |_|_|_|    1, 1, 1;
%e A228531 .        _ _ _ _
%e A228531 4  1    |    _ _|    4,
%e A228531 4  2    |  _|_ _|    2, 2,
%e A228531 4  3    | |  _ _|    1, 3,
%e A228531 4  4    | | |  _|    1, 1, 2,
%e A228531 4  5    |_|_|_|_|    1, 1, 1, 1;
%e A228531 .
%e A228531 Triangle begins:
%e A228531 [1];
%e A228531 [2],[1,1];
%e A228531 [3],[1,2],[1,1,1];
%e A228531 [4],[2,2],[1,3],[1,1,2],[1,1,1,1];
%e A228531 [5],[2,3],[1,4],[1,2,2],[1,1,3],[1,1,1,2],[1,1,1,1,1];
%e A228531 [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 A228531 [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 A228531 ...
%t A228531 revlexsort[f_,c_]:=OrderedQ[PadRight[{c,f}]];
%t A228531 Join@@Table[Sort[Reverse/@IntegerPartitions[n],revlexsort],{n,0,8}] (* _Gus Wiseman_, May 23 2020 *)
%Y A228531 Row lengths are A000041.
%Y A228531 Partition sums are A036042.
%Y A228531 Partition minima are A182715.
%Y A228531 Partition lengths are A333486.
%Y A228531 The lexicographic version (sum/lex) is A026791.
%Y A228531 Compositions under the same order (sum/revlex) are A066099.
%Y A228531 The colexicographic version (sum/colex) is A080576.
%Y A228531 The version for non-reversed partitions is A080577.
%Y A228531 The length-sensitive version (sum/length/revlex) is A334302.
%Y A228531 The Heinz numbers of these partitions are A334436.
%Y A228531 Partitions in colexicographic order (sum/colex) are A211992.
%Y A228531 Partitions in lexicographic order (sum/lex) are A193073.
%Y A228531 Cf. A026792, A036036, A049085, A103921, A112798, A115623, A129129, A228351, A331581, A334435, A334439, A334442.
%K A228531 nonn,tabf
%O A228531 1,2
%A A228531 _Omar E. Pol_, Aug 30 2013