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%I #46 Apr 01 2017 21:02:32
%S 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,
%T 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,
%U 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
%N 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.
%C Also 0 together the alternating sums of A220517.
%C 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.
%C 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:
%C .
%C . j Diagram 1 Partitions Diagram 2
%C . _ _ _ _ _ _ _ _ _ _ _ _ _ _
%C . 15 |_ _ _ _ | 7 _ _ _ _ |
%C . 14 |_ _ _ _|_ | 4+3 _ _ _ _|_ |
%C . 13 |_ _ _ | | 5+2 _ _ _ | |
%C . 12 |_ _ _|_ _|_ | 3+2+2 _ _ _|_ _|_ |
%C . 11 |_ _ _ | | 6+1 _ _ _ | |
%C . 10 |_ _ _|_ | | 3+3+1 _ _ _|_ | |
%C . 9 |_ _ | | | 4+2+1 _ _ | | |
%C . 8 |_ _|_ _|_ | | 2+2+2+1 _ _|_ _|_ | |
%C . 7 |_ _ _ | | | 5+1+1 _ _ _ | | |
%C . 6 |_ _ _|_ | | | 3+2+1+1 _ _ _|_ | | |
%C . 5 |_ _ | | | | 4+1+1+1 _ _ | | | |
%C . 4 |_ _|_ | | | | 2+2+1+1+1 _ _|_ | | | |
%C . 3 |_ _ | | | | | 3+1+1+1+1 _ _ | | | | |
%C . 2 |_ | | | | | | 2+1+1+1+1+1 _ | | | | | |
%C . 1 |_|_|_|_|_|_|_| 1+1+1+1+1+1+1 | | | | | | |
%C .
%C . 1 2 3 4 5 6 7
%C .
%C 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.
%C Also the diagram has the property that it can be transformed in a Dyck path (see example).
%C The height of the peaks and the valleys of the infinite Dyck path give this sequence.
%C Q: Is this Dyck path a fractal?
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa408.jpg">Visualization of regions in a diagram for A006128</a>
%F a(0) = 0; a(n) = a(n-1) + (-1)^(n-1)*A220517(n), n >= 1.
%e Illustration of initial terms (n = 0..21):
%e . 11
%e . /
%e . /
%e . /
%e . 7 /
%e . /\ 6 /
%e . 5 / \ 5 /\/
%e . /\ / \ /\ / 5
%e . 3 / \ 3 / \ / \/
%e . 2 /\ 2 / \ /\/ \ 2 / 3
%e . 1 /\ / \ /\/ \ / 2 \ /\/
%e . /\/ \/ \/ 1 \/ \/ 1
%e . 0 0 0 0 0 0
%e .
%e Note that the k-th largest peak between two valleys at height 0 is also A000041(k) and the next term is always 0.
%e .
%e Written as an irregular triangle in which row k has length 2*A187219(k), k >= 1, the sequence begins:
%e 0,1;
%e 0,2;
%e 0,3;
%e 0,2,1,5;
%e 0,3,2,7;
%e 0,2,1,5,3,6,5,11;
%e 0,3,2,7,5,9,8,15;
%e 0,2,1,5,3,6,5,11,7,12,11,15,14,22;
%e 0,3,2,7,5,9,8,15,11,14,13,19,17,22,21,30;
%e 0,2,1,5,3,6,5,11,7,12,11,15,14,22,15,19,18,25,23,29,28,33,32,42;
%e ...
%Y Column 1 is A000004. Right border gives A000041 for the positive integers.
%Y Cf. A006128, A135010, A138137, A139582, A141285, A186412, A187219, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610.
%K nonn,tabf
%O 0,4
%A _Omar E. Pol_, Nov 03 2013
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