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A036036 Triangle read by rows in which row n lists all the parts of all reversed partitions of n, sorted first by length and then lexicographically.

%I A036036 #80 May 16 2020 01:52:13
%S A036036 1,2,1,1,3,1,2,1,1,1,4,1,3,2,2,1,1,2,1,1,1,1,5,1,4,2,3,1,1,3,1,2,2,1,
%T A036036 1,1,2,1,1,1,1,1,6,1,5,2,4,3,3,1,1,4,1,2,3,2,2,2,1,1,1,3,1,1,2,2,1,1,
%U A036036 1,1,2,1,1,1,1,1,1,7,1,6,2,5,3,4,1,1,5,1,2,4,1,3,3,2,2,3,1,1,1
%N A036036 Triangle read by rows in which row n lists all the parts of all reversed partitions of n, sorted first by length and then lexicographically.
%C A036036 First differs from A334442 for reversed partitions of 9. Namely, this sequence has (1,4,4) before (2,2,5), while A334442 has (2,2,5) before (1,4,4). - _Gus Wiseman_, May 07 2020
%C A036036 This is the "Abramowitz and Stegun" ordering of the partitions, referenced in numerous other sequences. The partitions are in reverse order of the conjugates of the partitions in Mathematica order (A080577). Each partition is the conjugate of the corresponding partition in Maple order (A080576). - _Franklin T. Adams-Watters_, Oct 18 2006
%C A036036 The "Abramowitz and Stegun" ordering of the partitions is the graded reflected colexicographic ordering of the partitions. - _Daniel Forgues_, Jan 19 2011
%C A036036 The "Abramowitz and Stegun" ordering of partitions has been traced back to C. F. Hindenburg, 1779, in the Knuth reference, p. 38. See the Hindenburg link, pp. 77-5 with the listing of the partitions for n=10.  This is also mentioned in the P. Luschny link. - _Wolfdieter Lang_, Apr 04 2011
%C A036036 The "Abramowitz and Stegun" order used here means that the partitions of a given number are listed by increasing number of (nonzero) parts, then by increasing lexicographical order with parts in (weakly) indecreasing order. This differs from n=9 on from A334442 which considers reverse lexicographic order of parts in (weakly) decreasing order. - _M. F. Hasler_, Jul 12 2015, corrected thanks to _Gus Wiseman_, May 14 2020
%C A036036 This is the Abramowitz-Stegun ordering of reversed partitions (finite weakly increasing sequences of positive integers). The same ordering of non-reversed partitions is A334301. - _Gus Wiseman_, May 07 2020
%D A036036 Abramowitz and Stegun, Handbook, p. 831, column labeled "pi".
%D A036036 D. Knuth, The Art of Computer Programming, Vol. 4, fascicle 3, 7.2.1.4,  Addison-Wesley, 2005.
%H A036036 Franklin T. Adams-Watters, <a href="/A036036/b036036.txt">First 20 rows, flattened</a>.
%H A036036 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 A036036 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>.
%H A036036 C. F. Hindenburg, <a href="http://books.google.com/books?id=oIM_AAAAcAAJ">Infinitinomii Dignitatum Exponentis Indeterminati</a>, Goettingen 1779.
%H A036036 Peter Luschny, <a href="http://www.luschny.de/math/seq/CountingWithPartitions.html">Counting with partitions</a>.
%H A036036 OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions#A comparison">Orderings of partitions (a comparison)</a>.
%H A036036 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%e A036036 1
%e A036036 2; 1,1
%e A036036 3; 1,2; 1,1,1
%e A036036 4; 1,3; 2,2; 1,1,2; 1,1,1,1
%e A036036 5; 1,4; 2,3; 1,1,3; 1,2,2; 1,1,1,2; 1,1,1,1,1;
%e A036036 6; 1,5; 2,4; 3,3; 1,1,4; 1,2,3; 2,2,2; 1,1,1,3; 1,1,2,2; 1,1,1,1,2; 1,1,1,1,1,1;
%e A036036 ...
%t A036036 Join@@Table[Sort[Reverse/@IntegerPartitions[n]],{n,0,8}] (* _Gus Wiseman_, May 07 2020 *)
%t A036036 - or -
%t A036036 colen[f_,c_]:=OrderedQ[{Reverse[f],Reverse[c]}];
%t A036036 Reverse/@Join@@Table[Sort[IntegerPartitions[n],colen],{n,0,8}] (* _Gus Wiseman_, May 07 2020 *)
%o A036036 (PARI) T036036(n,k)=k&&return(T036036(n)[k]);concat(partitions(n))
%o A036036 \\ If 2nd arg "k" is not given, return the n-th row as a vector. Assumes PARI version >= 2.7.1. See A193073 for "hand made" code.
%o A036036 concat(vector(8,n,T036036(n))) \\ to get the "flattened" sequence
%o A036036 \\ _M. F. Hasler_, Jul 12 2015
%Y A036036 Cf. A036037-A036040.
%Y A036036 See A036037 for the graded colexicographic ordering.
%Y A036036 See A080576 for the Maple (graded reflected lexicographic) ordering.
%Y A036036 See A080577 for the Mathematica (graded reverse lexicographic) ordering.
%Y A036036 See A193073 for the graded lexicographic ordering.
%Y A036036 See A228100 for the Fenner-Loizou (binary tree) ordering.
%Y A036036 The version ignoring length is A026791.
%Y A036036 Same as A036037 with partitions reversed.
%Y A036036 The lengths of these partitions are A036043.
%Y A036036 The number of distinct parts is A103921.
%Y A036036 The corresponding ordering of compositions is A124734.
%Y A036036 Showing partitions as Heinz numbers gives A185974.
%Y A036036 The version for non-reversed partitions is A334301.
%Y A036036 Lexicographically ordered reversed partitions are A026791.
%Y A036036 Sorting reversed partitions by Heinz number gives A112798.
%Y A036036 The version for revlex instead of lex is A334302.
%Y A036036 The version for revlex instead of colex is A334442.
%Y A036036 Cf. A000041, A211992, A228531, A296774, A334433, A334435, A334436, A334437, A334438, A334439, A334440, A334441.
%K A036036 nonn,easy,nice,tabf,look
%O A036036 1,2
%A A036036 _N. J. A. Sloane_
%E A036036 Edited by _Daniel Forgues_, Jan 21 2011
%E A036036 Edited by _M. F. Hasler_, Jul 12 2015
%E A036036 Name corrected by _Gus Wiseman_, May 12 2020