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A229946
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Height of the peaks and the valleys in the Dyck path whose j-th ascending line segment has A141285(j) steps and whose j-th descending line segment has A194446(j) steps.
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3
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0, 1, 0, 2, 0, 3, 0, 2, 1, 5, 0, 3, 2, 7, 0, 2, 1, 5, 3, 6, 5, 11, 0, 3, 2, 7, 5, 9, 8, 15, 0, 2, 1, 5, 3, 6, 5, 11, 7, 12, 11, 15, 14, 22, 0, 3, 2, 7, 5, 9, 8, 15, 11, 14, 13, 19, 17, 22, 21, 30, 0, 2, 1, 5, 3, 6, 5, 11, 7, 12, 11, 15, 14, 22, 15, 19, 18, 25, 23, 29, 28, 33, 32, 42, 0
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OFFSET
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0,4
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COMMENTS
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Also 0 together the alternating sums of A220517.
The master diagram of regions of the set of partitions of all positive integers is a total dissection of the first quadrant of the square grid in which the j-th horizontal line segments has length A141285(j) and the j-th vertical line segment has length A194446(j). For the definition of "region" see A206437. The first A000041(k) regions of the diagram represent the set of partitions of k in colexicographic order (see A211992). The length of the j-th horizontal line segment equals the largest part of the j-th partition of k and equals the largest part of the j-th region of the diagram. The length of the j-th vertical line segment (which is the line segment ending in row j) equals the number of parts in the j-th region.
For k = 7, the diagram 1 represents the partitions of 7. The diagram 2 is a minimalist version of the structure which does not contain the axes [X, Y]. See below:
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. j Diagram 1 Partitions Diagram 2
. _ _ _ _ _ _ _ _ _ _ _ _ _ _
. 15 |_ _ _ _ | 7 _ _ _ _ |
. 14 |_ _ _ _|_ | 4+3 _ _ _ _|_ |
. 13 |_ _ _ | | 5+2 _ _ _ | |
. 12 |_ _ _|_ _|_ | 3+2+2 _ _ _|_ _|_ |
. 11 |_ _ _ | | 6+1 _ _ _ | |
. 10 |_ _ _|_ | | 3+3+1 _ _ _|_ | |
. 9 |_ _ | | | 4+2+1 _ _ | | |
. 8 |_ _|_ _|_ | | 2+2+2+1 _ _|_ _|_ | |
. 7 |_ _ _ | | | 5+1+1 _ _ _ | | |
. 6 |_ _ _|_ | | | 3+2+1+1 _ _ _|_ | | |
. 5 |_ _ | | | | 4+1+1+1 _ _ | | | |
. 4 |_ _|_ | | | | 2+2+1+1+1 _ _|_ | | | |
. 3 |_ _ | | | | | 3+1+1+1+1 _ _ | | | | |
. 2 |_ | | | | | | 2+1+1+1+1+1 _ | | | | | |
. 1 |_|_|_|_|_|_|_| 1+1+1+1+1+1+1 | | | | | | |
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. 1 2 3 4 5 6 7
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The second diagram has the property that if the number of regions is also the number of partitions of k so the sum of the lengths of all horizontal line segment equals the sum of the lengths of all vertical line segments and equals A006128(k), for k >= 1.
Also the diagram has the property that it can be transformed in a Dyck path (see example).
The height of the peaks and the valleys of the infinite Dyck path give this sequence.
Q: Is this Dyck path a fractal?
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LINKS
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FORMULA
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a(0) = 0; a(n) = a(n-1) + (-1)^(n-1)*A220517(n), n >= 1.
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EXAMPLE
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Illustration of initial terms (n = 0..21):
. 11
. /
. /
. /
. 7 /
. /\ 6 /
. 5 / \ 5 /\/
. /\ / \ /\ / 5
. 3 / \ 3 / \ / \/
. 2 /\ 2 / \ /\/ \ 2 / 3
. 1 /\ / \ /\/ \ / 2 \ /\/
. /\/ \/ \/ 1 \/ \/ 1
. 0 0 0 0 0 0
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Note that the k-th largest peak between two valleys at height 0 is also A000041(k) and the next term is always 0.
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Written as an irregular triangle in which row k has length 2*A187219(k), k >= 1, the sequence begins:
0,1;
0,2;
0,3;
0,2,1,5;
0,3,2,7;
0,2,1,5,3,6,5,11;
0,3,2,7,5,9,8,15;
0,2,1,5,3,6,5,11,7,12,11,15,14,22;
0,3,2,7,5,9,8,15,11,14,13,19,17,22,21,30;
0,2,1,5,3,6,5,11,7,12,11,15,14,22,15,19,18,25,23,29,28,33,32,42;
...
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CROSSREFS
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Column 1 is A000004. Right border gives A000041 for the positive integers.
Cf. A006128, A135010, A138137, A139582, A141285, A186412, A187219, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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