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A228347
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Triangle of regions and compositions of the positive integers (see Comments lines for definition).
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3
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1, 1, 2, 0, 0, 1, 1, 1, 2, 3, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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1,3
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COMMENTS
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Triangle read by rows in which row n lists A129760(n) zeros followed by the A006519(n) elements of the row A001511(n) of triangle A090996, n >= 1.
The equivalent sequence for partitions is A186114.
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LINKS
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EXAMPLE
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----------------------------------------------------------
. 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
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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 column are the parts of the n-th composition of k, with compositions in colexicographic order.
Triangle begins:
1;
1,2;
0,0,1;
1,1,2,3;
0,0,0,0,1;
0,0,0,0,1,2;
0,0,0,0,0,0,1;
1,1,1,1,2,2,3,4;
0,0,0,0,0,0,0,0,1;
0,0,0,0,0,0,0,0,1,2;
0,0,0,0,0,0,0,0,0,0,1;
0,0,0,0,0,0,0,0,1,1,2,3;
0,0,0,0,0,0,0,0,0,0,0,0,1;
0,0,0,0,0,0,0,0,0,0,0,0,1,2;
0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;
...
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CROSSREFS
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Cf. A001792, A001787, A006519, A011782, A065120, A096996, A129760, A186114, A187816, A187818, A206437, A228350, A228351, A228366, A228367, A228370, A228371, A228525, A228526.
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KEYWORD
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AUTHOR
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STATUS
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approved
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