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A187818
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Triangle read by rows in which row n lists the first 2^(n-1) terms of A038712 in nonincreasing order, n >= 1.
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6
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1, 3, 1, 7, 3, 1, 1, 15, 7, 3, 3, 1, 1, 1, 1, 31, 15, 7, 7, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 63, 31, 15, 15, 7, 7, 7, 7, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 127, 63, 31, 31, 15, 15, 15, 15, 7, 7, 7, 7, 7, 7, 7, 7, 3, 3
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OFFSET
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1,2
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COMMENTS
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T(n,k) is also the sum of all parts of the k-th largest region of the diagram of regions of the set of compositions of n, n >= 1, k >= 1, see example.
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LINKS
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EXAMPLE
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For n = 5 the diagram of regions of the set of compositions of 5 has 2^(5-1) regions, see below:
------------------------------------------------------
. a tree of sum Diagram
Region of all parts of regions Composition
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. _ _ _ _ _
1 | 1 | |_| | | | | 1, 1, 1, 1, 1
2 | 3 | |_ _| | | | 2, 1, 1, 1
3 | 1 | |_| | | | 1, 2, 1, 1
4 | 7 | |_ _ _| | | 3, 1, 1
5 | 1 | |_| | | | 1, 1, 2, 1
6 | 3 | |_ _| | | 2, 2, 1
7 | 1 | |_| | | 1, 3, 1
8 | 15 | |_ _ _ _| | 4, 1
9 | 1 | |_| | | | 1, 1, 1, 2
10 | 3 | |_ _| | | 2, 1, 2
11 | 1 | |_| | | 1, 2, 2
12 | 7 | |_ _ _| | 3, 2
13 | 1 | |_| | | 1, 1, 3
14 | 3 | |_ _| | 2, 3
15 | 1 | |_| | 1, 4
16 | 31 | |_ _ _ _ _| 5
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The first largest region in the diagram is the 16th region which contains 16 parts and the sum of parts is 31, so T(5,1) = 31. The second largest region is the 8th region which contains 8 parts and the sum of parts is 15, so T(5,2) = 15. The third and the fourth largest regions are both the 4th region and the 12th region, each contains 4 parts and the sum of parts is 7, so T(5,3) = 7 and T(5,4) = 7. And so on. The sequence of the sum of all parts of the k-th largest region of the diagram is [31, 15, 7, 7, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1], the same as the 5th row of triangle, as shown below.
Triangle begins:
1;
3,1;
7,3,1,1;
15,7,3,3,1,1,1,1;
31,15,7,7,3,3,3,3,1,1,1,1,1,1,1,1;
63,31,15,15,7,7,7,7,3,3,3,3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
...
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CROSSREFS
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Cf. A000079, A000225, A001511, A001787, A001792, A006519, A011782, A038712, A065120, A187816, A228525, A228369.
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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STATUS
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approved
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