A228348 Triangle of regions and compositions of the positive integers (see Comments lines for definition).

1, 2, 1, 1, 0, 0, 3, 2, 1, 1, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 4, 3, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Omar E. Pol, Aug 21 2013

Keywords

• nonn
• tabl

Comments

Triangle read by rows in which row n lists the A006519(n) elements of the row A001511(n) of triangle A065120 followed by A129760(n) zeros, n >= 1.

The equivalent sequence for integer partitions is A193870.

Examples

			----------------------------------------------------------
.             Diagram                Triangle
Compositions    of            of compositions (rows)
of 5          regions          and regions (columns)
----------------------------------------------------------
.            _ _ _ _ _
5           |_        |                                 5
1+4         |_|_      |                               1 4
2+3         |_  |     |                             2 0 3
1+1+3       |_|_|_    |                           1 1 0 3
3+2         |_    |   |                         3 0 0 0 2
1+2+2       |_|_  |   |                       1 2 0 0 0 2
2+1+2       |_  | |   |                     2 0 1 0 0 0 2
1+1+1+2     |_|_|_|_  |                   1 1 0 1 0 0 0 2
4+1         |_      | |                 4 0 0 0 0 0 0 0 1
1+3+1       |_|_    | |               1 3 0 0 0 0 0 0 0 1
2+2+1       |_  |   | |             2 0 2 0 0 0 0 0 0 0 1
1+1+2+1     |_|_|_  | |           1 1 0 2 0 0 0 0 0 0 0 1
3+1+1       |_    | | |         3 0 0 0 1 0 0 0 0 0 0 0 1
1+2+1+1     |_|_  | | |       1 2 0 0 0 1 0 0 0 0 0 0 0 1
2+1+1+1     |_  | | | |     2 0 1 0 0 0 1 0 0 0 0 0 0 0 1
1+1+1+1+1   |_|_|_|_|_|   1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1
.
For the positive integer k consider the first 2^(k-1) rows of triangle, as shown below. The positive terms of the n-th row are the parts of the n-th region of the diagram of regions of the set of compositions of k. The positive terms of the n-th diagonal are the parts of the n-th composition of k, with compositions in colexicographic order.
Triangle begins:
1;
2,1;
1,0,0;
3,2,1,1;
1,0,0,0,0;
2,1,0,0,0,0;
1,0,0,0,0,0,0;
4,3,2,2,1,1,1,1;
1,0,0,0,0,0,0,0,0;
2,1,0,0,0,0,0,0,0,0;
1,0,0,0,0,0,0,0,0,0,0;
3,2,1,1,0,0,0,0,0,0,0,0;
1,0,0,0,0,0,0,0,0,0,0,0,0;
2,1,0,0,0,0,0,0,0,0,0,0,0,0;
1,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1;
...
		

Crossrefs

Mirror of A228347. Column 1 is A001511. Right border gives A036987. Also right border gives A209229, n >= 1. Positive terms give A228350.

Cf. A001792, A001787, A006519, A011782, A065120, A129760, A187816, A187818, A193870, A206437, A228349, A228351, A228366, A228367, A228370, A228371, A228525, A228526.