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Triangle of regions and compositions of the positive integers (see Comments lines for definition).
3

%I #15 Oct 19 2013 03:20:17

%S 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,

%T 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,

%U 1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0

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

%C 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.

%C The equivalent sequence for integer partitions is A193870.

%e ----------------------------------------------------------

%e . Diagram Triangle

%e Compositions of of compositions (rows)

%e of 5 regions and regions (columns)

%e ----------------------------------------------------------

%e . _ _ _ _ _

%e 5 |_ | 5

%e 1+4 |_|_ | 1 4

%e 2+3 |_ | | 2 0 3

%e 1+1+3 |_|_|_ | 1 1 0 3

%e 3+2 |_ | | 3 0 0 0 2

%e 1+2+2 |_|_ | | 1 2 0 0 0 2

%e 2+1+2 |_ | | | 2 0 1 0 0 0 2

%e 1+1+1+2 |_|_|_|_ | 1 1 0 1 0 0 0 2

%e 4+1 |_ | | 4 0 0 0 0 0 0 0 1

%e 1+3+1 |_|_ | | 1 3 0 0 0 0 0 0 0 1

%e 2+2+1 |_ | | | 2 0 2 0 0 0 0 0 0 0 1

%e 1+1+2+1 |_|_|_ | | 1 1 0 2 0 0 0 0 0 0 0 1

%e 3+1+1 |_ | | | 3 0 0 0 1 0 0 0 0 0 0 0 1

%e 1+2+1+1 |_|_ | | | 1 2 0 0 0 1 0 0 0 0 0 0 0 1

%e 2+1+1+1 |_ | | | | 2 0 1 0 0 0 1 0 0 0 0 0 0 0 1

%e 1+1+1+1+1 |_|_|_|_|_| 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1

%e .

%e 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.

%e Triangle begins:

%e 1;

%e 2,1;

%e 1,0,0;

%e 3,2,1,1;

%e 1,0,0,0,0;

%e 2,1,0,0,0,0;

%e 1,0,0,0,0,0,0;

%e 4,3,2,2,1,1,1,1;

%e 1,0,0,0,0,0,0,0,0;

%e 2,1,0,0,0,0,0,0,0,0;

%e 1,0,0,0,0,0,0,0,0,0,0;

%e 3,2,1,1,0,0,0,0,0,0,0,0;

%e 1,0,0,0,0,0,0,0,0,0,0,0,0;

%e 2,1,0,0,0,0,0,0,0,0,0,0,0,0;

%e 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0;

%e 5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1;

%e ...

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

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

%K nonn,tabl

%O 1,2

%A _Omar E. Pol_, Aug 21 2013