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A334439 Irregular triangle whose rows are all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

%I A334439 #15 Sep 22 2023 05:18:21
%S A334439 1,2,1,1,3,2,1,1,1,1,4,3,1,2,2,2,1,1,1,1,1,1,5,4,1,3,2,3,1,1,2,2,1,2,
%T A334439 1,1,1,1,1,1,1,1,6,5,1,4,2,3,3,4,1,1,3,2,1,2,2,2,3,1,1,1,2,2,1,1,2,1,
%U A334439 1,1,1,1,1,1,1,1,1,7,6,1,5,2,4,3,5,1,1
%N A334439 Irregular triangle whose rows are all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.
%C A334439 First differs from A036037 for partitions of 9. Namely, this sequence has (5,2,2) before (4,4,1), while A036037 has (4,4,1) before (5,2,2).
%C A334439 This is the Abramowitz-Stegun ordering of integer partitions (A334301) except that the finer order is reverse-lexicographic instead of lexicographic. The version for reversed partitions is A334302.
%H A334439 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%e A334439 The sequence of all partitions begins:
%e A334439   ()      (32)     (21111)   (22111)    (4211)      (63)
%e A334439   (1)     (311)    (111111)  (211111)   (3311)      (54)
%e A334439   (2)     (221)    (7)       (1111111)  (3221)      (711)
%e A334439   (11)    (2111)   (61)      (8)        (2222)      (621)
%e A334439   (3)     (11111)  (52)      (71)       (41111)     (531)
%e A334439   (21)    (6)      (43)      (62)       (32111)     (522)
%e A334439   (111)   (51)     (511)     (53)       (22211)     (441)
%e A334439   (4)     (42)     (421)     (44)       (311111)    (432)
%e A334439   (31)    (33)     (331)     (611)      (221111)    (333)
%e A334439   (22)    (411)    (322)     (521)      (2111111)   (6111)
%e A334439   (211)   (321)    (4111)    (431)      (11111111)  (5211)
%e A334439   (1111)  (222)    (3211)    (422)      (9)         (4311)
%e A334439   (5)     (3111)   (2221)    (332)      (81)        (4221)
%e A334439   (41)    (2211)   (31111)   (5111)     (72)        (3321)
%e A334439 This sequence can also be interpreted as the following triangle, whose n-th row is itself a finite triangle with A000041(n) rows.
%e A334439                   0
%e A334439                  (1)
%e A334439                (2)(11)
%e A334439              (3)(21)(111)
%e A334439         (4)(31)(22)(211)(1111)
%e A334439   (5)(41)(32)(311)(221)(2111)(11111)
%e A334439 Showing partitions as their Heinz numbers (see A334438) gives:
%e A334439    1
%e A334439    2
%e A334439    3   4
%e A334439    5   6   8
%e A334439    7  10   9  12  16
%e A334439   11  14  15  20  18  24  32
%e A334439   13  22  21  25  28  30  27  40  36  48  64
%e A334439   17  26  33  35  44  42  50  45  56  60  54  80  72  96 128
%t A334439 revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]<Length[c],OrderedQ[{c,f}]];
%t A334439 Join@@Table[Sort[IntegerPartitions[n],revlensort],{n,0,8}]
%Y A334439 The version for colex instead of revlex is A036037.
%Y A334439 Row lengths are A036043.
%Y A334439 Ignoring length gives A080577.
%Y A334439 Number of distinct elements in row n appears to be A103921(n).
%Y A334439 The version for compositions is A296774.
%Y A334439 The Abramowitz-Stegun version (sum/length/lex) is A334301.
%Y A334439 The version for reversed partitions is A334302.
%Y A334439 Taking Heinz numbers gives A334438.
%Y A334439 The version with partitions reversed is A334442.
%Y A334439 Lexicographically ordered reversed partitions are A026791.
%Y A334439 Lexicographically ordered partitions are A193073.
%Y A334439 Sorting partitions by Heinz number gives A296150.
%Y A334439 Cf. A000041, A036036, A112798, A124734, A129129, A185974, A228100, A228531, A334433, A334435, A334436.
%K A334439 nonn,tabf
%O A334439 0,2
%A A334439 _Gus Wiseman_, May 03 2020