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A211992 Triangle read by rows in which row n lists the partitions of n in colexicographic order. 32
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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].

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

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

LINKS

Joerg Arndt, Table of n, a(n) for n = 1..10000

OEIS Wiki, Orderings of partitions

EXAMPLE

From Omar E. Pol, Aug 24 2013: (Start)

Illustration of initial terms:

-----------------------------------------

n      Diagram          Partition

-----------------------------------------

.       _

1      |_|              1;

.       _ _

2      |_| |            1, 1,

2      |_ _|            2;

.       _ _ _

3      |_| | |          1, 1, 1,

3      |_ _| |          2, 1,

3      |_ _ _|          3;

.       _ _ _ _

4      |_| | | |        1, 1, 1, 1,

4      |_ _| | |        2, 1, 1,

4      |_ _ _| |        3, 1,

4      |_ _|   |        2, 2,

4      |_ _ _ _|        4;

.       _ _ _ _ _

5      |_| | | | |      1, 1, 1, 1, 1,

5      |_ _| | | |      2, 1, 1, 1,

5      |_ _ _| | |      3, 1, 1,

5      |_ _|   | |      2, 2, 1,

5      |_ _ _ _| |      4, 1,

5      |_ _ _|   |      3, 2,

5      |_ _ _ _ _|      5;

.       _ _ _ _ _ _

6      |_| | | | | |    1, 1, 1, 1, 1, 1,

6      |_ _| | | | |    2, 1, 1, 1, 1,

6      |_ _ _| | | |    3, 1, 1, 1,

6      |_ _|   | | |    2, 2, 1, 1,

6      |_ _ _ _| | |    4, 1, 1,

6      |_ _ _|   | |    3, 2, 1,

6      |_ _ _ _ _| |    5, 1,

6      |_ _|   |   |    2, 2, 2,

6      |_ _ _ _|   |    4, 2,

6      |_ _ _|     |    3, 3,

6      |_ _ _ _ _ _|    6;

...

Triangle begins:

[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], [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], [5,1], [2,2,2], [4,2], [3,3], [6];

(End)

PROG

(PARI)

gen_part(n)=

{  /* Generate partitions of n as weakly increasing lists (order is lex): */

    my(ct = 0);

    my(m, pt);

    my(x, y);

    \\ init:

    my( a = vector( n + (n<=1) ) );

    a[1] = 0;  a[2] = n;  m = 2;

    while ( m!=1,

        y = a[m] - 1;

        m -= 1;

        x = a[m] + 1;

        while ( x<=y,

            a[m] = x;

            y = y - x;

            m += 1;

        );

        a[m] = x + y;

        pt = vector(m, j, a[j]);

    /* for A026791 print partition: */

\\        for (j=1, m, print1(pt[j], ", ") );

    /* for A211992 print partition as weakly decreasing list (order is colex): */

        forstep (j=m, 1, -1, print1(pt[j], ", ") );

        ct += 1;

    );

    return(ct);

}

for(n=1, 10, gen_part(n) );

\\ Joerg Arndt, Sep 02 2013

CROSSREFS

Cf. A026791, A026792, A141285, A194446, A228525, A228531.

Sequence in context: A093993 A193073 A228100 * A182937 A185147 A206921

Adjacent sequences:  A211989 A211990 A211991 * A211993 A211994 A211995

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Aug 18 2012

STATUS

approved

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Last modified June 19 00:55 EDT 2019. Contains 324217 sequences. (Running on oeis4.)