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A187816
Triangle read by rows in which row n lists the first 2^(n-1) terms of A006519 in nonincreasing order, n >= 1.
6
1, 2, 1, 4, 2, 1, 1, 8, 4, 2, 2, 1, 1, 1, 1, 16, 8, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 32, 16, 8, 8, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 64, 32, 16, 16, 8, 8, 8, 8, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2
OFFSET
1,2
COMMENTS
T(n,k) is also the number of parts in the k-th largest region of the diagram of regions of the set of compositions of n, n >= 1, k >= 1, see example.
Row lengths is A000079.
Row sums give A001792(n-1).
EXAMPLE
For n = 5 the diagram of regions of the set of compositions of 5 has 2^(5-1) regions, see below:
------------------------------------------------------
. as a tree
. of number Diagram
Region of parts of regions Composition
------------------------------------------------------
. _ _ _ _ _
1 | 1 | |_| | | | | 1, 1, 1, 1, 1
2 | 2 | |_ _| | | | 2, 1, 1, 1
3 | 1 | |_| | | | 1, 2, 1, 1
4 | 4 | |_ _ _| | | 3, 1, 1
5 | 1 | |_| | | | 1, 1, 2, 1
6 | 2 | |_ _| | | 2, 2, 1
7 | 1 | |_| | | 1, 3, 1
8 | 8 | |_ _ _ _| | 4, 1
9 | 1 | |_| | | | 1, 1, 1, 2
10 | 2 | |_ _| | | 2, 1, 2
11 | 1 | |_| | | 1, 2, 2
12 | 4 | |_ _ _| | 3, 2
13 | 1 | |_| | | 1, 1, 3
14 | 2 | |_ _| | 2, 3
15 | 1 | |_| | 1, 4
16 | 16 | |_ _ _ _ _| 5
.
The first largest region in the diagram is the 16th region which contains 16 parts, so T(5,1) = 16. The second largest region is the 8th region which contains 8 parts, so T(5,2) = 8. The third and the fourth largest regions are both the 4th region and the 12th region, each contains 4 parts, so T(5,3) = 4 and T(5,4) = 4. And so on. The sequence of the number of parts of the k-th largest region of the diagram is [16, 8, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1], the same as the 5th row of triangle, as shown below.
Triangle begins:
1;
2,1;
4,2,1,1;
8,4,2,2,1,1,1,1;
16,8,4,4,2,2,2,2,1,1,1,1,1,1,1,1;
32,16,8,8,4,4,4,4,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
...
KEYWORD
nonn,tabf,easy
AUTHOR
Omar E. Pol, Sep 10 2013
STATUS
approved