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A080577 Triangle in which n-th row lists all partitions of n, in graded reverse lexicographic ordering.

%I A080577 #75 Jul 28 2025 10:06:10
%S A080577 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 A080577 1,1,1,1,1,1,1,1,6,5,1,4,2,4,1,1,3,3,3,2,1,3,1,1,1,2,2,2,2,2,1,1,2,1,
%U A080577 1,1,1,1,1,1,1,1,1,7,6,1,5,2,5,1,1,4,3,4,2,1,4,1,1,1,3,3,1,3,2
%N A080577 Triangle in which n-th row lists all partitions of n, in graded reverse lexicographic ordering.
%C A080577 This is the "Mathematica" ordering of the partitions, referenced in numerous other sequences. The partitions of each integer are in reverse order of the conjugates of the partitions in Abramowitz and Stegun order (A036036). They are in the reverse of the order of the partitions in Maple order (A080576). - _Franklin T. Adams-Watters_, Oct 18 2006
%C A080577 The graded reverse lexicographic ordering of the partitions is often referred to as the "Canonical" ordering of the partitions. - _Daniel Forgues_, Jan 21 2011
%C A080577 Also the "MAGMA" ordering of the partitions. - _Jason Kimberley_, Oct 28 2011
%C A080577 Also an intuitive ordering described but not formalized in [Hardy and Wright] the first four editions of which precede [Abramowitz and Stegun]. - _L. Edson Jeffery_, Aug 03 2013
%C A080577 Also the "Sage" ordering of the partitions. - _Peter Luschny_, Aug 12 2013
%C A080577 While this is the order used for the constructive function "IntegerPartitions", it is different from Mathematica's canonical ordering of finite expressions, the latter giving A036036 if parts of partitions are read in reversed (weakly increasing) order, or A334301 if in the usual (weakly decreasing) order. - _Gus Wiseman_, May 08 2020
%D A080577 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Oxford, Fifth edition, 1979, p. 273.
%D A080577 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 287.
%H A080577 Franklin T. Adams-Watters, <a href="/A080577/b080577.txt">First 20 rows, flattened</a>
%H A080577 M. Abramowitz and I. A. Stegun, eds., <a href="https://personal.math.ubc.ca/~cbm/aands/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 831.
%H A080577 OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions#A_comparison">Orderings of partitions (a comparison)</a>.
%H A080577 Sergei Viznyuk, <a href="http://phystech.com/ftp/s_A209936.c">C Program</a>
%H A080577 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%e A080577 First five rows are:
%e A080577   {{1}}
%e A080577   {{2}, {1, 1}}
%e A080577   {{3}, {2, 1}, {1, 1, 1}}
%e A080577   {{4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}}
%e A080577   {{5}, {4, 1}, {3, 2}, {3, 1, 1}, {2, 2, 1}, {2, 1, 1, 1}, {1, 1, 1, 1, 1}}
%e A080577 Up to the fifth row, this is exactly the same as the colexicographic ordering A036037. The first row which differs is the sixth one, which reads ((6), (5,1), (4,2), (4,1,1), (3,3), (3,2,1), (3,1,1,1), (2,2,2), (2,2,1,1), (2,1,1,1,1), (1,1,1,1,1,1)). - _M. F. Hasler_, Jan 23 2020
%e A080577 From _Gus Wiseman_, May 08 2020: (Start)
%e A080577 The sequence of all partitions begins:
%e A080577   ()         (3,2)        (2,1,1,1,1)    (2,2,1,1,1)
%e A080577   (1)        (3,1,1)      (1,1,1,1,1,1)  (2,1,1,1,1,1)
%e A080577   (2)        (2,2,1)      (7)            (1,1,1,1,1,1,1)
%e A080577   (1,1)      (2,1,1,1)    (6,1)          (8)
%e A080577   (3)        (1,1,1,1,1)  (5,2)          (7,1)
%e A080577   (2,1)      (6)          (5,1,1)        (6,2)
%e A080577   (1,1,1)    (5,1)        (4,3)          (6,1,1)
%e A080577   (4)        (4,2)        (4,2,1)        (5,3)
%e A080577   (3,1)      (4,1,1)      (4,1,1,1)      (5,2,1)
%e A080577   (2,2)      (3,3)        (3,3,1)        (5,1,1,1)
%e A080577   (2,1,1)    (3,2,1)      (3,2,2)        (4,4)
%e A080577   (1,1,1,1)  (3,1,1,1)    (3,2,1,1)      (4,3,1)
%e A080577   (5)        (2,2,2)      (3,1,1,1,1)    (4,2,2)
%e A080577   (4,1)      (2,2,1,1)    (2,2,2,1)      (4,2,1,1)
%e A080577 The triangle with partitions shown as Heinz numbers (A129129) begins:
%e A080577    1
%e A080577    2
%e A080577    3   4
%e A080577    5   6   8
%e A080577    7  10   9  12  16
%e A080577   11  14  15  20  18  24  32
%e A080577   13  22  21  28  25  30  40  27  36  48  64
%e A080577   17  26  33  44  35  42  56  50  45  60  80  54  72  96 128
%e A080577 (End)
%p A080577 b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
%p A080577     [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
%p A080577 T:= n-> map(x-> x[], b(n$2))[]:
%p A080577 seq(T(n), n=1..8);  # _Alois P. Heinz_, Jan 29 2020
%t A080577 <<DiscreteMath`Combinatorica`; Partition[6]
%t A080577 (* Or, from version 6 on : *) Table[ IntegerPartitions[n], {n, 1, 7}] // Flatten (* _Jean-François Alcover_, Dec 10 2012 *)
%t A080577 revlexsort[f_,c_]:=OrderedQ[PadRight[{c,f}]];
%t A080577 Join@@Table[Sort[IntegerPartitions[n],revlexsort],{n,0,8}] (* _Gus Wiseman_, May 08 2020 *)
%o A080577 (Magma) &cat[&cat Partitions(n):n in[1..7]]; // _Jason Kimberley_, Oct 28 2011
%o A080577 (Sage)
%o A080577 L = []
%o A080577 for n in range(8): L += list(Partitions(n))
%o A080577 flatten(L)   # _Peter Luschny_, Aug 12 2013
%o A080577 (PARI) A080577_row(n)={vecsort(apply(t->Vecrev(t),partitions(n)),,4)} \\ _M. F. Hasler_, Jan 21 2020
%Y A080577 See A080576 Maple (graded reflected lexicographic) ordering.
%Y A080577 See A036036 for the Hindenburg (graded reflected colexicographic) ordering (listed in the Abramowitz and Stegun Handbook).
%Y A080577 See A036037 for graded colexicographic ordering.
%Y A080577 See A228100 for the Fenner-Loizou (binary tree) ordering.
%Y A080577 Differs from A036037 at a(48).
%Y A080577 See A322761 for a compressed version.
%Y A080577 Lexicographically ordered reversed partitions are A026791.
%Y A080577 Reverse-colexicographically ordered partitions are A026792.
%Y A080577 Compositions under this ordering are A066099.
%Y A080577 Distinct parts of these partitions are counted by A115623.
%Y A080577 Taking Heinz numbers gives A129129.
%Y A080577 Lexicographically ordered partitions are A193073.
%Y A080577 Colexicographically ordered partitions are A211992.
%Y A080577 Reading partitions in reverse (weakly increasing) order gives A228531.
%Y A080577 Lengths of these partitions are A238966.
%Y A080577 Sorting partitions by Heinz number gives A296150.
%Y A080577 The maxima of these partitions are A331581.
%Y A080577 The length-sensitive version is A334439.
%Y A080577 Cf. A000041, A048793, A063008, A185974, A334301, A334434, A334436, A334438.
%K A080577 nonn,tabf
%O A080577 1,2
%A A080577 _N. J. A. Sloane_, Mar 23 2003