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