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A036037 Triangle read by rows in which row n lists all the parts of all the partitions of n, sorted first by length and then colexicographically.

%I A036037 #54 Jun 05 2020 08:21:23
%S A036037 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 A036037 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 A036037 1,1,1,1,1,1,1,1,1,7,6,1,5,2,4,3,5,1,1,4,2,1,3,3,1,3,2,2,4,1,1
%N A036037 Triangle read by rows in which row n lists all the parts of all the partitions of n, sorted first by length and then colexicographically.
%C A036037 First differs from A334439 for partitions of 9. Namely, this sequence has (4,4,1) before (5,2,2), while A334439 has (5,2,2) before (4,4,1). - _Gus Wiseman_, May 08 2020
%C A036037 This is also a list of all the possible prime signatures of a number, arranged in graded colexicographic ordering. - _N. J. A. Sloane_, Feb 09 2014
%C A036037 This is also the Abramowitz-Stegun ordering of reversed partitions (A036036) if the partitions are reversed again after sorting. Partitions sorted first by sum and then colexicographically are A211992. - _Gus Wiseman_, May 08 2020
%H A036037 Robert Price, <a href="/A036037/b036037.txt">Table of n, a(n) for n = 1..3615, 15 rows.</a>
%H A036037 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%e A036037 First five rows are:
%e A036037 {{1}}
%e A036037 {{2}, {1, 1}}
%e A036037 {{3}, {2, 1}, {1, 1, 1}}
%e A036037 {{4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}}
%e A036037 {{5}, {4, 1}, {3, 2}, {3, 1, 1}, {2, 2, 1}, {2, 1, 1, 1}, {1, 1, 1, 1, 1}}
%e A036037 Up to the fifth row, this is exactly the same as the reverse lexicographic ordering A080577. The first row which differs is the sixth one, which reads ((6), (5,1), (4,2), (3,3), (4,1,1), (3,2,1), (2,2,2), (3,1,1,1), (2,2,1,1), (2,1,1,1,1), (1,1,1,1,1,1)). - _M. F. Hasler_, Jan 23 2020
%e A036037 From _Gus Wiseman_, May 08 2020: (Start)
%e A036037 The sequence of all partitions begins:
%e A036037   ()         (3,2)        (2,1,1,1,1)
%e A036037   (1)        (3,1,1)      (1,1,1,1,1,1)
%e A036037   (2)        (2,2,1)      (7)
%e A036037   (1,1)      (2,1,1,1)    (6,1)
%e A036037   (3)        (1,1,1,1,1)  (5,2)
%e A036037   (2,1)      (6)          (4,3)
%e A036037   (1,1,1)    (5,1)        (5,1,1)
%e A036037   (4)        (4,2)        (4,2,1)
%e A036037   (3,1)      (3,3)        (3,3,1)
%e A036037   (2,2)      (4,1,1)      (3,2,2)
%e A036037   (2,1,1)    (3,2,1)      (4,1,1,1)
%e A036037   (1,1,1,1)  (2,2,2)      (3,2,1,1)
%e A036037   (5)        (3,1,1,1)    (2,2,2,1)
%e A036037   (4,1)      (2,2,1,1)    (3,1,1,1,1)
%e A036037 (End)
%t A036037 Reverse/@Join@@Table[Sort[Reverse/@IntegerPartitions[n]],{n,8}] (* _Gus Wiseman_, May 08 2020 *)
%t A036037 - or -
%t A036037 colen[f_,c_]:=OrderedQ[{Reverse[f],Reverse[c]}];
%t A036037 Join@@Table[Sort[IntegerPartitions[n],colen],{n,8}] (* _Gus Wiseman_, May 08 2020 *)
%Y A036037 See A036036 for the graded reflected colexicographic ("Abramowitz and Stegun" or Hindenburg) ordering.
%Y A036037 See A080576 for the graded reflected lexicographic ("Maple") ordering.
%Y A036037 See A080577 for the graded reverse lexicographic ("Mathematica") ordering: differs from a(48) on!
%Y A036037 See A228100 for the Fenner-Loizou (binary tree) ordering.
%Y A036037 See also A036038, A036039, A036040: (multinomial coefficients).
%Y A036037 Partition lengths are A036043.
%Y A036037 Reversing all partitions gives A036036.
%Y A036037 The number of distinct parts is A103921.
%Y A036037 Taking Heinz numbers gives A185974.
%Y A036037 The version ignoring length is A211992.
%Y A036037 The version for revlex instead of colex is A334439.
%Y A036037 Lexicographically ordered reversed partitions are A026791.
%Y A036037 Reverse-lexicographically ordered partitions are A080577.
%Y A036037 Sorting partitions by Heinz number gives A296150.
%Y A036037 Cf. A000041, A124734, A193073, A228100, A228531, A296774, A334301, A334433, A334436, A334437, A334442.
%K A036037 nonn,easy,tabf
%O A036037 1,2
%A A036037 _N. J. A. Sloane_
%E A036037 Name corrected by _Gus Wiseman_, May 12 2020
%E A036037 Mathematica programs corrected to reflect offset of one and not zero by _Robert Price_, Jun 04 2020