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A229946 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.

%I A229946 #46 Apr 01 2017 21:02:32
%S A229946 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 A229946 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 A229946 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 A229946 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 A229946 Also 0 together the alternating sums of A220517.
%C A229946 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 A229946 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 A229946 .
%C A229946 .  j     Diagram 1         Partitions           Diagram 2
%C A229946 .      _ _ _ _ _ _ _                          _ _ _ _ _ _ _
%C A229946 . 15  |_ _ _ _      |      7                  _ _ _ _      |
%C A229946 . 14  |_ _ _ _|_    |      4+3                _ _ _ _|_    |
%C A229946 . 13  |_ _ _    |   |      5+2                _ _ _    |   |
%C A229946 . 12  |_ _ _|_ _|_  |      3+2+2              _ _ _|_ _|_  |
%C A229946 . 11  |_ _ _      | |      6+1                _ _ _      | |
%C A229946 . 10  |_ _ _|_    | |      3+3+1              _ _ _|_    | |
%C A229946 .  9  |_ _    |   | |      4+2+1              _ _    |   | |
%C A229946 .  8  |_ _|_ _|_  | |      2+2+2+1            _ _|_ _|_  | |
%C A229946 .  7  |_ _ _    | | |      5+1+1              _ _ _    | | |
%C A229946 .  6  |_ _ _|_  | | |      3+2+1+1            _ _ _|_  | | |
%C A229946 .  5  |_ _    | | | |      4+1+1+1            _ _    | | | |
%C A229946 .  4  |_ _|_  | | | |      2+2+1+1+1          _ _|_  | | | |
%C A229946 .  3  |_ _  | | | | |      3+1+1+1+1          _ _  | | | | |
%C A229946 .  2  |_  | | | | | |      2+1+1+1+1+1        _  | | | | | |
%C A229946 .  1  |_|_|_|_|_|_|_|      1+1+1+1+1+1+1       | | | | | | |
%C A229946 .
%C A229946 .      1 2 3 4 5 6 7
%C A229946 .
%C A229946 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 A229946 Also the diagram has the property that it can be transformed in a Dyck path (see example).
%C A229946 The height of the peaks and the valleys of the infinite Dyck path give this sequence.
%C A229946 Q: Is this Dyck path a fractal?
%H A229946 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa408.jpg">Visualization of regions in a diagram for A006128</a>
%F A229946 a(0) = 0; a(n) = a(n-1) + (-1)^(n-1)*A220517(n), n >= 1.
%e A229946 Illustration of initial terms (n = 0..21):
%e A229946 .                                                             11
%e A229946 .                                                             /
%e A229946 .                                                            /
%e A229946 .                                                           /
%e A229946 .                                   7                      /
%e A229946 .                                   /\                 6  /
%e A229946 .                     5            /  \           5    /\/
%e A229946 .                     /\          /    \          /\  / 5
%e A229946 .           3        /  \     3  /      \        /  \/
%e A229946 .      2    /\   2  /    \    /\/        \   2  /   3
%e A229946 .   1  /\  /  \  /\/      \  / 2          \  /\/
%e A229946 .   /\/  \/    \/ 1        \/              \/ 1
%e A229946 .  0 0   0     0           0               0
%e A229946 .
%e A229946 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 A229946 .
%e A229946 Written as an irregular triangle in which row k has length 2*A187219(k), k >= 1, the sequence begins:
%e A229946 0,1;
%e A229946 0,2;
%e A229946 0,3;
%e A229946 0,2,1,5;
%e A229946 0,3,2,7;
%e A229946 0,2,1,5,3,6,5,11;
%e A229946 0,3,2,7,5,9,8,15;
%e A229946 0,2,1,5,3,6,5,11,7,12,11,15,14,22;
%e A229946 0,3,2,7,5,9,8,15,11,14,13,19,17,22,21,30;
%e A229946 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 A229946 ...
%Y A229946 Column 1 is A000004. Right border gives A000041 for the positive integers.
%Y A229946 Cf. A006128, A135010, A138137, A139582, A141285, A186412, A187219, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610.
%K A229946 nonn,tabf
%O A229946 0,4
%A A229946 _Omar E. Pol_, Nov 03 2013