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A334302 Irregular triangle read by rows where row k is the k-th reversed integer partition, if reversed partitions are sorted first by sum, then by length, and finally reverse-lexicographically.

%I A334302 #12 May 28 2020 05:03:05
%S A334302 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 A334302 1,1,2,1,1,1,1,1,6,3,3,2,4,1,5,2,2,2,1,2,3,1,1,4,1,1,2,2,1,1,1,3,1,1,
%U A334302 1,1,2,1,1,1,1,1,1,7,3,4,2,5,1,6,2,2,3
%N A334302 Irregular triangle read by rows where row k is the k-th reversed integer partition, if reversed partitions are sorted first by sum, then by length, and finally reverse-lexicographically.
%H A334302 OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>
%H A334302 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%e A334302 The sequence of all reversed partitions begins:
%e A334302   ()         (1,4)        (1,1,1,1,2)
%e A334302   (1)        (1,2,2)      (1,1,1,1,1,1)
%e A334302   (2)        (1,1,3)      (7)
%e A334302   (1,1)      (1,1,1,2)    (3,4)
%e A334302   (3)        (1,1,1,1,1)  (2,5)
%e A334302   (1,2)      (6)          (1,6)
%e A334302   (1,1,1)    (3,3)        (2,2,3)
%e A334302   (4)        (2,4)        (1,3,3)
%e A334302   (2,2)      (1,5)        (1,2,4)
%e A334302   (1,3)      (2,2,2)      (1,1,5)
%e A334302   (1,1,2)    (1,2,3)      (1,2,2,2)
%e A334302   (1,1,1,1)  (1,1,4)      (1,1,2,3)
%e A334302   (5)        (1,1,2,2)    (1,1,1,4)
%e A334302   (2,3)      (1,1,1,3)    (1,1,1,2,2)
%e A334302 This sequence can also be interpreted as the following triangle, whose n-th row is itself a finite triangle with A000041(n) rows.
%e A334302                             0
%e A334302                            (1)
%e A334302                         (2) (1,1)
%e A334302                     (3) (1,2) (1,1,1)
%e A334302             (4) (2,2) (1,3) (1,1,2) (1,1,1,1)
%e A334302   (5) (2,3) (1,4) (1,2,2) (1,1,3) (1,1,1,2) (1,1,1,1,1)
%e A334302 Showing partitions as their Heinz numbers (see A334435) gives:
%e A334302    1
%e A334302    2
%e A334302    3   4
%e A334302    5   6   8
%e A334302    7   9  10  12  16
%e A334302   11  15  14  18  20  24  32
%e A334302   13  25  21  22  27  30  28  36  40  48  64
%e A334302   17  35  33  26  45  50  42  44  54  60  56  72  80  96 128
%t A334302 revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]<Length[c],OrderedQ[{c,f}]];
%t A334302 Join@@Table[Sort[Sort/@IntegerPartitions[n],revlensort],{n,0,8}]
%Y A334302 Row lengths are A036043.
%Y A334302 Lexicographically ordered reversed partitions are A026791.
%Y A334302 The dual ordering (sum/length/lex) of reversed partitions is A036036.
%Y A334302 Reverse-lexicographically ordered partitions are A080577.
%Y A334302 Sorting reversed partitions by Heinz number gives A112798.
%Y A334302 Lexicographically ordered partitions are A193073.
%Y A334302 Graded Heinz numbers are A215366.
%Y A334302 Ignoring length gives A228531.
%Y A334302 Sorting partitions by Heinz number gives A296150.
%Y A334302 The version for compositions is A296774.
%Y A334302 The dual ordering (sum/length/lex) of non-reversed partitions is A334301.
%Y A334302 Taking Heinz numbers gives A334435.
%Y A334302 The version for regular (non-reversed) partitions is A334439 (not A036037).
%Y A334302 Cf. A000041, A048793, A066099, A080576, A124734, A162247, A211992, A228100, A228351.
%K A334302 nonn,tabf
%O A334302 0,2
%A A334302 _Gus Wiseman_, Apr 30 2020