cp's OEIS Frontend

A211992 Triangle read by rows in which row n lists the partitions of n in colexicographic order.

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