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%I #10 Sep 22 2023 09:01:40
%S 1,2,1,2,3,1,3,4,2,3,1,4,5,1,2,3,2,4,1,5,6,1,2,4,3,4,2,5,1,6,7,1,3,4,
%T 1,2,5,3,5,2,6,1,7,8,2,3,4,1,3,5,4,5,1,2,6,3,6,2,7,1,8,9,1,2,3,4,2,3,
%U 5,1,4,5,1,3,6,4,6,1,2,7,3,7,2,8,1,9,10
%N Flattened tetrangle of reversed strict integer partitions sorted first by sum, then colexicographically.
%C The zeroth row contains only the empty partition.
%C A tetrangle is a sequence of finite triangles.
%H Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%e Tetrangle begins:
%e 0: ()
%e 1: (1)
%e 2: (2)
%e 3: (12)(3)
%e 4: (13)(4)
%e 5: (23)(14)(5)
%e 6: (123)(24)(15)(6)
%e 7: (124)(34)(25)(16)(7)
%e 8: (134)(125)(35)(26)(17)(8)
%e 9: (234)(135)(45)(126)(36)(27)(18)(9)
%t colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
%t Table[Sort[Reverse/@Select[IntegerPartitions[n],UnsameQ@@#&],colex],{n,0,10}]
%Y Positions of first appearances are A015724.
%Y Triangle sums are A066189.
%Y The non-strict version is A080576.
%Y Taking lex instead of colex gives A246688 (non-reversed: A344086).
%Y The non-reversed version is A344087.
%Y Taking revlex instead of colex gives A344089 (non-reversed: A118457).
%Y A026793 gives reversed strict partitions in A-S order (sum/length/lex).
%Y A319247 sorts strict partitions by Heinz number.
%Y A329631 sorts reversed strict partitions by Heinz number.
%Y A344090 gives strict partitions in A-S order (sum/length/lex).
%Y Cf. A005117, A014466, A209862.
%Y Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A272020, A299755, A296774, A304038, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344091.
%Y Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.
%K nonn,tabf
%O 0,2
%A _Gus Wiseman_, May 12 2021