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
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Aug 24 2013
STATUS
approved