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A228368
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Difference between the n-th element of the ruler function and the highest power of 2 dividing n.
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2
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0, 0, 0, -1, 0, 0, 0, -4, 0, 0, 0, -1, 0, 0, 0, -11, 0, 0, 0, -1, 0, 0, 0, -4, 0, 0, 0, -1, 0, 0, 0, -26, 0, 0, 0, -1, 0, 0, 0, -4, 0, 0, 0, -1, 0, 0, 0, -11, 0, 0, 0, -1, 0, 0, 0, -4, 0, 0, 0, -1, 0, 0, 0, -57, 0, 0, 0, -1, 0, 0, 0, -4, 0, 0, 0, -1, 0, 0, 0, -11, 0, 0, 0, -1, 0, 0, 0, -4, 0, 0, 0, -1, 0, 0, 0, -26
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OFFSET
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1,8
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COMMENTS
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Also rank of the n-th region of the diagram of compositions of j, if 1 <= n <= 2^(j-1), see example.
Here the rank of a region is defined as the largest part minus the number of parts (similar to the Dyson's rank of a partition).
The equivalent sequence for integer partitions is A194447.
Also triangle read by rows in which T(j,k) is the rank of the k-th region of the j-th section of the set of compositions in colexicographic order of any integer >= j. See A228366.
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LINKS
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FORMULA
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EXAMPLE
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Illustration of initial terms (n = 1..16):
-----------------------------------------------
. Largest Number of
. Diagram of part of parts of
. compositions region n region n
-----------------------------------------------
-----------------------------------------------
.
1 _| | | | | 1 1 0
2 _ _| | | | 2 2 0
3 _| | | | 1 1 0
4 _ _ _| | | 3 4 -1
5 _| | | | 1 1 0
6 _ _| | | 2 2 0
7 _| | | 1 1 0
8 _ _ _ _| | 4 8 -4
9 _| | | | 1 1 0
10 _ _| | | 2 2 0
11 _| | | 1 1 0
12 _ _ _| | 3 4 -1
13 _| | | 1 1 0
14 _ _| | 2 2 0
15 _| | 1 1 0
16 _ _ _ _ _| 5 16 -11
.
Written as an array read by rows with four columns the first three columns contain only zeros.
0, 0, 0, -1;
0, 0, 0, -4;
0, 0, 0, -1;
0, 0, 0, -11;
0, 0, 0, -1;
0, 0, 0, -4;
0, 0, 0, -1;
0, 0, 0, -26;
...
Written as a triangle T(j,k) the sequence begins:
0;
0;
0,-1;
0,0,0,-4;
0,0,0,-1,0,0,0,-11;
0,0,0,-1,0,0,0,-4,0,0,0,-1,0,0,0,-26;
0,0,0,-1,0,0,0,-4,0,0,0,-1,0,0,0,-11,0,0,0,-1,0,0,0,-4,0, 0,0,-1,0,0,0,-57;
...
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PROG
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(Python)
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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