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%I #38 May 12 2020 12:58:52
%S 1,1,1,2,1,1,1,2,1,3,1,1,1,1,2,1,1,3,1,2,2,4,1,1,1,1,1,2,1,1,1,3,1,1,
%T 2,2,1,4,1,3,2,5,1,1,1,1,1,1,2,1,1,1,1,3,1,1,1,2,2,1,1,4,1,1,3,2,1,5,
%U 1,2,2,2,4,2,3,3,6,1,1,1,1,1,1,1,2,1,1,1,1,1,3,1,1,1,1,2,2,1,1,1,4,1,1,1,3,2,1,1,5,1,1,2,2,2,1,4,2,1,3,3,1,6,1,3,2,2,5,2,4,3,7
%N Triangle read by rows in which row n lists the partitions of n in colexicographic order.
%C The order of the partitions of every integer is reversed with respect to A026792. For example: in A026792 the partitions of 3 are listed as [3], [2, 1], [1, 1, 1], however here the partitions of 3 are listed as [1, 1, 1], [2, 1], [3].
%C Row n has length A006128(n). Row sums give A066186. Right border gives A000027. The equivalent sequence for compositions (ordered partitions) is A228525. - _Omar E. Pol_, Aug 24 2013
%C The representation of the partitions (for fixed n) is as (weakly) decreasing lists of parts, the order between individual partitions (for the same n) is co-lexicographic. The equivalent sequence for partitions as (weakly) increasing lists and lexicographic order is A026791. - _Joerg Arndt_, Sep 02 2013
%H Joerg Arndt, <a href="/A211992/b211992.txt">Table of n, a(n) for n = 1..10000</a>
%H OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>
%H Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%e From _Omar E. Pol_, Aug 24 2013: (Start)
%e Illustration of initial terms:
%e -----------------------------------------
%e n Diagram Partition
%e -----------------------------------------
%e . _
%e 1 |_| 1;
%e . _ _
%e 2 |_| | 1, 1,
%e 2 |_ _| 2;
%e . _ _ _
%e 3 |_| | | 1, 1, 1,
%e 3 |_ _| | 2, 1,
%e 3 |_ _ _| 3;
%e . _ _ _ _
%e 4 |_| | | | 1, 1, 1, 1,
%e 4 |_ _| | | 2, 1, 1,
%e 4 |_ _ _| | 3, 1,
%e 4 |_ _| | 2, 2,
%e 4 |_ _ _ _| 4;
%e . _ _ _ _ _
%e 5 |_| | | | | 1, 1, 1, 1, 1,
%e 5 |_ _| | | | 2, 1, 1, 1,
%e 5 |_ _ _| | | 3, 1, 1,
%e 5 |_ _| | | 2, 2, 1,
%e 5 |_ _ _ _| | 4, 1,
%e 5 |_ _ _| | 3, 2,
%e 5 |_ _ _ _ _| 5;
%e . _ _ _ _ _ _
%e 6 |_| | | | | | 1, 1, 1, 1, 1, 1,
%e 6 |_ _| | | | | 2, 1, 1, 1, 1,
%e 6 |_ _ _| | | | 3, 1, 1, 1,
%e 6 |_ _| | | | 2, 2, 1, 1,
%e 6 |_ _ _ _| | | 4, 1, 1,
%e 6 |_ _ _| | | 3, 2, 1,
%e 6 |_ _ _ _ _| | 5, 1,
%e 6 |_ _| | | 2, 2, 2,
%e 6 |_ _ _ _| | 4, 2,
%e 6 |_ _ _| | 3, 3,
%e 6 |_ _ _ _ _ _| 6;
%e ...
%e Triangle begins:
%e [1];
%e [1,1], [2];
%e [1,1,1], [2,1], [3];
%e [1,1,1,1], [2,1,1], [3,1], [2,2], [4];
%e [1,1,1,1,1], [2,1,1,1], [3,1,1], [2,2,1], [4,1], [3,2], [5];
%e [1,1,1,1,1,1], [2,1,1,1,1], [3,1,1,1], [2,2,1,1], [4,1,1], [3,2,1], [5,1], [2,2,2], [4,2], [3,3], [6];
%e (End)
%e From _Gus Wiseman_, May 10 2020: (Start)
%e The triangle with partitions shown as Heinz numbers (A334437) begins:
%e 1
%e 2
%e 4 3
%e 8 6 5
%e 16 12 10 9 7
%e 32 24 20 18 14 15 11
%e 64 48 40 36 28 30 22 27 21 25 13
%e 128 96 80 72 56 60 44 54 42 50 26 45 33 35 17
%e (End)
%t colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
%t Join@@Table[Sort[IntegerPartitions[n],colex],{n,0,6}] (* _Gus Wiseman_, May 10 2020 *)
%o (PARI)
%o gen_part(n)=
%o { /* Generate partitions of n as weakly increasing lists (order is lex): */
%o my(ct = 0);
%o my(m, pt);
%o my(x, y);
%o \\ init:
%o my( a = vector( n + (n<=1) ) );
%o a[1] = 0; a[2] = n; m = 2;
%o while ( m!=1,
%o y = a[m] - 1;
%o m -= 1;
%o x = a[m] + 1;
%o while ( x<=y,
%o a[m] = x;
%o y = y - x;
%o m += 1;
%o );
%o a[m] = x + y;
%o pt = vector(m, j, a[j]);
%o /* for A026791 print partition: */
%o \\ for (j=1, m, print1(pt[j],", ") );
%o /* for A211992 print partition as weakly decreasing list (order is colex): */
%o forstep (j=m, 1, -1, print1(pt[j],", ") );
%o ct += 1;
%o );
%o return(ct);
%o }
%o for(n=1, 10, gen_part(n) );
%o \\ _Joerg Arndt_, Sep 02 2013
%Y Cf. A026791, A141285, A194446, A228531.
%Y The graded reversed version is A026792.
%Y The length-sensitive refinement is A036037.
%Y The version for reversed partitions is A080576.
%Y Partition lengths are A193173.
%Y Partition maxima are A194546.
%Y Partition minima are A196931.
%Y The version for compositions is A228525.
%Y The Heinz numbers of these partitions are A334437.
%Y Cf. A036036, A080577, A193073, A228100, A296150, A331581, A334301, A334302, A334436, A334439, A334442.
%K nonn,tabf
%O 1,4
%A _Omar E. Pol_, Aug 18 2012
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