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A334442 Irregular triangle whose reversed rows are all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

%I A334442 #20 Sep 22 2023 08:46:46
%S A334442 1,2,1,1,3,1,2,1,1,1,4,1,3,2,2,1,1,2,1,1,1,1,5,1,4,2,3,1,1,3,1,2,2,1,
%T A334442 1,1,2,1,1,1,1,1,6,1,5,2,4,3,3,1,1,4,1,2,3,2,2,2,1,1,1,3,1,1,2,2,1,1,
%U A334442 1,1,2,1,1,1,1,1,1,7,1,6,2,5,3,4,1,1,5
%N A334442 Irregular triangle whose reversed rows are all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.
%C A334442 First differs from A036036 for reversed partitions of 9. Namely, this sequence has (2,2,5) before (1,4,4), while A036036 has (1,4,4) before (2,2,5).
%H A334442 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%e A334442 The sequence of all partitions begins:
%e A334442   ()         (2,3)        (1,1,1,1,2)    (1,1,1,2,2)
%e A334442   (1)        (1,1,3)      (1,1,1,1,1,1)  (1,1,1,1,1,2)
%e A334442   (2)        (1,2,2)      (7)            (1,1,1,1,1,1,1)
%e A334442   (1,1)      (1,1,1,2)    (1,6)          (8)
%e A334442   (3)        (1,1,1,1,1)  (2,5)          (1,7)
%e A334442   (1,2)      (6)          (3,4)          (2,6)
%e A334442   (1,1,1)    (1,5)        (1,1,5)        (3,5)
%e A334442   (4)        (2,4)        (1,2,4)        (4,4)
%e A334442   (1,3)      (3,3)        (1,3,3)        (1,1,6)
%e A334442   (2,2)      (1,1,4)      (2,2,3)        (1,2,5)
%e A334442   (1,1,2)    (1,2,3)      (1,1,1,4)      (1,3,4)
%e A334442   (1,1,1,1)  (2,2,2)      (1,1,2,3)      (2,2,4)
%e A334442   (5)        (1,1,1,3)    (1,2,2,2)      (2,3,3)
%e A334442   (1,4)      (1,1,2,2)    (1,1,1,1,3)    (1,1,1,5)
%e A334442 This sequence can also be interpreted as the following triangle:
%e A334442                   0
%e A334442                  (1)
%e A334442                (2)(11)
%e A334442              (3)(12)(111)
%e A334442         (4)(13)(22)(112)(1111)
%e A334442   (5)(14)(23)(113)(122)(1112)(11111)
%e A334442 Taking Heinz numbers (A334438) gives:
%e A334442    1
%e A334442    2
%e A334442    3   4
%e A334442    5   6   8
%e A334442    7  10   9  12  16
%e A334442   11  14  15  20  18  24  32
%e A334442   13  22  21  25  28  30  27  40  36  48  64
%e A334442   17  26  33  35  44  42  50  45  56  60  54  80  72  96 128
%t A334442 revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]<Length[c],OrderedQ[{c,f}]];
%t A334442 Join@@Reverse/@Join@@Table[Sort[IntegerPartitions[n],revlensort],{n,0,8}]
%o A334442 (PARI) A334442_row(n)=vecsort(partitions(n),p->concat(#p,-Vecrev(p))) \\ Rows of triangle defined in EXAMPLE (all partitions of n). Wrap into [Vec(p)|p<-...] to avoid "Vecsmall". - _M. F. Hasler_, May 14 2020
%Y A334442 Row lengths are A036043.
%Y A334442 The version for reversed partitions is A334301.
%Y A334442 The version for colex instead of revlex is A334302.
%Y A334442 Taking Heinz numbers gives A334438.
%Y A334442 The version with rows reversed is A334439.
%Y A334442 Ignoring length gives A335122.
%Y A334442 Lexicographically ordered reversed partitions are A026791.
%Y A334442 Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
%Y A334442 Partitions in increasing-length colex order (sum/length/colex) are A036037.
%Y A334442 Reverse-lexicographically ordered partitions are A080577.
%Y A334442 Lexicographically ordered partitions are A193073.
%Y A334442 Partitions in colexicographic order (sum/colex) are A211992.
%Y A334442 Sorting partitions by Heinz number gives A296150.
%Y A334442 Cf. A026791, A112798, A124734, A129129, A185974, A228100, A228531, A296774, A334433, A334435, A334436.
%K A334442 nonn,tabf
%O A334442 0,2
%A A334442 _Gus Wiseman_, May 07 2020