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%I #52 Sep 22 2023 07:57:05
%S 1,1,1,2,1,1,1,1,2,3,1,1,1,1,1,1,2,2,2,1,3,4,1,1,1,1,1,1,1,1,2,1,2,2,
%T 1,1,3,2,3,1,4,5,1,1,1,1,1,1,1,1,1,1,2,1,1,2,2,2,2,2,1,1,1,3,1,2,3,3,
%U 3,1,1,4,2,4,1,5,6,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,2,2,1,2,2,2
%N Triangle in which n-th row lists all partitions of n, in graded reflected lexicographic order.
%C The graded reflected lexicographic ordering of the partitions is used by Maple. - _Daniel Forgues_, Jan 19 2011
%C Each partition here is the conjugate of the corresponding partition in Abramowitz and Stegun order (A036036). The partitions are in the reverse of the order of the partitions in Mathematica order (A080577). - _Franklin T. Adams-Watters_, Oct 18 2006
%C Reversing all partitions gives A193073 (the non-reflected version). The version for reversed (weakly increasing) partitions is A211992. Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036. - _Gus Wiseman_, May 20 2020
%C Also reversed integer partitions in colexicographic order, cf. A228531. - _Gus Wiseman_, May 31 2020
%H Alois P. Heinz, <a href="/A080576/b080576.txt">Rows n = 1..20, flattened</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. (uses Flash)
%H A. M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>
%H Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%e First five rows are:
%e [[1]]
%e [[1, 1], [2]]
%e [[1, 1, 1], [1, 2], [3]]
%e [[1, 1, 1, 1], [1, 1, 2], [2, 2], [1, 3], [4]]
%e [[1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 2, 2], [1, 1, 3], [2, 3], [1, 4], [5]]
%e From _Gus Wiseman_, May 20 2020: (Start)
%e The sequence of all reversed partitions begins:
%e () (122) (15) (25)
%e (1) (113) (6) (16)
%e (11) (23) (1111111) (7)
%e (2) (14) (111112) (11111111)
%e (111) (5) (11122) (1111112)
%e (12) (111111) (1222) (111122)
%e (3) (11112) (11113) (11222)
%e (1111) (1122) (1123) (2222)
%e (112) (222) (223) (111113)
%e (22) (1113) (133) (11123)
%e (13) (123) (1114) (1223)
%e (4) (33) (124) (1133)
%e (11111) (114) (34) (233)
%e (1112) (24) (115) (11114)
%e (End)
%p with(combinat); partition(6);
%t row[n_] := Flatten[Reverse /@ Reverse[SplitBy[Reverse /@ IntegerPartitions[n], Length]], 1]; Array[row, 7] // Flatten (* _Jean-François Alcover_, Dec 05 2016 *)
%t lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
%t Reverse/@Join@@Table[Sort[IntegerPartitions[n],lexsort],{n,0,8}] (* _Gus Wiseman_, May 20 2020 *)
%Y See A080577 for the Mathematica (graded reverse lexicographic) ordering.
%Y See A036036 for the Hindenburg (graded reflected colexicographic) ordering (listed in the Abramowitz and Stegun Handbook).
%Y See A036037 for the graded colexicographic ordering.
%Y See A193073 for the graded lexicographic ordering. - _M. F. Hasler_, Jul 16 2011
%Y See A228100 for the Fenner-Loizou (binary tree) ordering.
%Y Row n has A000041(n) partitions.
%Y Taking colexicographic instead of lexicographic gives A026791.
%Y Lengths of these partitions appear to be A049085.
%Y Reversing all partitions gives A193073 (the non-reflected version).
%Y The version for reversed (weakly increasing) partitions is A211992.
%Y The generalization to compositions is A228525.
%Y The Heinz numbers of these partitions are A334434.
%Y Cf. A026791, A036037, A112798, A129129, A185974, A228351, A228531, A334301, A334302, A334433, A334437.
%K nonn,tabf
%O 1,4
%A _N. J. A. Sloane_, Mar 23 2003
%E Edited by _Daniel Forgues_, Jan 21 2011
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