A228525 Triangle read by rows in which row n lists the compositions (ordered partitions) of n in colexicographic order.

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 2, 2, 1, 3, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 1, 4, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 3, 2, 1, 1, 3, 2, 3, 1, 4, 5, 1, 1, 1, 1, 1, 1

Offset

1,4

Comments

The representation of the compositions (for fixed n) is as lists of parts, the order between individual compositions (for the same n) is co-lexicographic. [Joerg Arndt, Sep 02 2013]

The equivalent sequence for partitions is A211992.

Row n has length A001792(n-1).

Row sums give A001787, n >= 1.

Links

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

Example

Illustration of initial terms:
---------------------------------
n  j     Diagram     Composition
---------------------------------
.         _
1  1     |_|         1;
.         _ _
2  1     |_| |       1, 1,
2  2     |_ _|       2;
.         _ _ _
3  1     |_| | |     1, 1, 1,
3  2     |_ _| |     2, 1,
3  3     |_|   |     1, 2,
3  4     |_ _ _|     3;
.         _ _ _ _
4  1     |_| | | |   1, 1, 1, 1,
4  2     |_ _| | |   2, 1, 1,
4  3     |_|   | |   1, 2, 1,
4  4     |_ _ _| |   3, 1,
4  5     |_| |   |   1, 1, 2,
4  6     |_ _|   |   2, 2,
4  7     |_|     |   1, 3,
4  8     |_ _ _ _|   4;
.
Triangle begins:
[1];
[1,1],[2];
[1,1,1],[2,1],[1,2],[3];
[1,1,1,1],[2,1,1],[1,2,1],[3,1],[1,1,2],[2,2],[1,3],[4];
[1,1,1,1,1],[2,1,1,1],[1,2,1,1],[3,1,1],[1,1,2,1],[2,2,1],[1,3,1],[4,1],[1,1,1,2],[2,1,2],[1,2,2],[3,2],[1,1,3],[2,3],[1,4],[5];

Prog

(PARI)
gen_comp(n)=
{  /* Generate compositions of n as lists of parts (order is lex): */
    my(ct = 0);
    my(m, z, pt);
    \\ init:
    my( a = vector(n, j, 1) );
    m = n;
    while ( 1,
        ct += 1;
        pt = vector(m, j, a[j]);
\\        /* for A228369  print composition: */
\\        for (j=1, m, print1(pt[j], ", ") );
        /* for A228525 print reversed (order is colex): */
        forstep (j=m, 1, -1, print1(pt[j], ", ") );
        if ( m<=1,  return(ct) );  \\ current is last
        a[m-1] += 1;
        z = a[m] - 2;
        a[m] = 1;
        m += z;
    );
    return(ct);
}
for(n=1, 12, gen_comp(n) );
\\ Joerg Arndt, Sep 02 2013

Crossrefs

Cf. A066099, A211992, A228351, A228369.

Sequence in context: A144379 A337260 A296772 * A352823 A335234 A353854

Adjacent sequences: A228522 A228523 A228524 * A228526 A228527 A228528

Keyword

nonn,tabf

Author

Omar E. Pol, Aug 24 2013

Status

approved