login
Triangle read by rows in which row n lists the partitions of n in colexicographic order.
77

%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