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%I #67 Sep 08 2022 08:45:09
%S 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 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 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 Triangle in which n-th row lists all partitions of n, in graded reverse lexicographic ordering.
%C 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 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 Also the "MAGMA" ordering of the partitions. - _Jason Kimberley_, Oct 28 2011
%C 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 Also the "Sage" ordering of the partitions. - _Peter Luschny_, Aug 12 2013
%C 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 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Oxford, Fifth edition, 1979, p. 273.
%H Franklin T. Adams-Watters, <a href="/A080577/b080577.txt">First 20 rows, flattened</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://people.math.sfu.ca/~cbm/aands/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 831.
%H OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions#A_comparison">Orderings of partitions (a comparison)</a>.
%H Sergei Viznyuk, <a href="http://phystech.com/ftp/s_A209936.c">C Program</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 {{2}, {1, 1}}
%e {{3}, {2, 1}, {1, 1, 1}}
%e {{4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}}
%e {{5}, {4, 1}, {3, 2}, {3, 1, 1}, {2, 2, 1}, {2, 1, 1, 1}, {1, 1, 1, 1, 1}}
%e 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 From _Gus Wiseman_, May 08 2020: (Start)
%e The sequence of all partitions begins:
%e () (3,2) (2,1,1,1,1) (2,2,1,1,1)
%e (1) (3,1,1) (1,1,1,1,1,1) (2,1,1,1,1,1)
%e (2) (2,2,1) (7) (1,1,1,1,1,1,1)
%e (1,1) (2,1,1,1) (6,1) (8)
%e (3) (1,1,1,1,1) (5,2) (7,1)
%e (2,1) (6) (5,1,1) (6,2)
%e (1,1,1) (5,1) (4,3) (6,1,1)
%e (4) (4,2) (4,2,1) (5,3)
%e (3,1) (4,1,1) (4,1,1,1) (5,2,1)
%e (2,2) (3,3) (3,3,1) (5,1,1,1)
%e (2,1,1) (3,2,1) (3,2,2) (4,4)
%e (1,1,1,1) (3,1,1,1) (3,2,1,1) (4,3,1)
%e (5) (2,2,2) (3,1,1,1,1) (4,2,2)
%e (4,1) (2,2,1,1) (2,2,2,1) (4,2,1,1)
%e The triangle with partitions shown as Heinz numbers (A129129) begins:
%e 1
%e 2
%e 3 4
%e 5 6 8
%e 7 10 9 12 16
%e 11 14 15 20 18 24 32
%e 13 22 21 28 25 30 40 27 36 48 64
%e 17 26 33 44 35 42 56 50 45 60 80 54 72 96 128
%e (End)
%p b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
%p [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
%p T:= n-> map(x-> x[], b(n$2))[]:
%p seq(T(n), n=1..8); # _Alois P. Heinz_, Jan 29 2020
%t <<DiscreteMath`Combinatorica`; Partition[6]
%t (* Or, from version 6 on : *) Table[ IntegerPartitions[n], {n, 1, 7}] // Flatten (* _Jean-François Alcover_, Dec 10 2012 *)
%t revlexsort[f_,c_]:=OrderedQ[PadRight[{c,f}]];
%t Join@@Table[Sort[IntegerPartitions[n],revlexsort],{n,0,8}] (* _Gus Wiseman_, May 08 2020 *)
%o (Magma) &cat[&cat Partitions(n):n in[1..7]]; // _Jason Kimberley_, Oct 28 2011
%o (Sage)
%o L = []
%o for n in range(8): L += list(Partitions(n))
%o flatten(L) # _Peter Luschny_, Aug 12 2013
%o (PARI) A080577_row(n)={vecsort(apply(t->Vecrev(t),partitions(n)),,4)} \\ _M. F. Hasler_, Jan 21 2020
%Y See A080576 Maple (graded reflected lexicographic) ordering.
%Y See A036036 for the Hindenburg (graded reflected colexicographic) ordering (listed in the Abramowitz and Stegun Handbook).
%Y See A036037 for graded colexicographic ordering.
%Y See A228100 for the Fenner-Loizou (binary tree) ordering.
%Y Differs from A036037 at a(48).
%Y See A322761 for a compressed version.
%Y Lexicographically ordered reversed partitions are A026791.
%Y Reverse-colexicographically ordered partitions are A026792.
%Y Compositions under this ordering are A066099.
%Y Distinct parts of these partitions are counted by A115623.
%Y Taking Heinz numbers gives A129129.
%Y Lexicographically ordered partitions are A193073.
%Y Colexicographically ordered partitions are A211992.
%Y Reading partitions in reverse (weakly increasing) order gives A228531.
%Y Lengths of these partitions are A238966.
%Y Sorting partitions by Heinz number gives A296150.
%Y The maxima of these partitions are A331581.
%Y The length-sensitive version is A334439.
%Y Cf. A000041, A048793, A063008, A185974, A334301, A334434, A334436, A334438.
%K nonn,tabf
%O 1,2
%A _N. J. A. Sloane_, Mar 23 2003
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