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%I #45 Apr 01 2017 21:02:02
%S 0,1,0,1,2,1,0,1,2,3,2,1,0,1,2,1,2,3,4,5,4,3,2,1,0,1,2,3,2,3,4,5,6,7,
%T 6,5,4,3,2,1,0,1,2,1,2,3,4,5,4,3,4,5,6,5,6,7,8,9,10,11,10,9,8,7,6,5,4,
%U 3,2,1,0,1,2,3,2,3,4,5,6,7,6,5,6,7,8,9,8,9,10,11,12,13,14,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1,0
%N Height after n-th step of the infinite Dyck path in which the k-th ascending line segment has A141285(k) steps and the k-th descending line segment has A194446(k) steps, n >= 0, k >= 1.
%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 sequence gives the height of the infinite Dyck path after n-th step.
%C The absolute values of the first differences give A000012.
%C For the height of the peaks and the valleys in the infinite Dyck path see A229946.
%C Q: Is this infinite 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>
%e Illustration of initial terms (n = 1..59):
%e .
%e 11 ...........................................................
%e . /
%e . /
%e . /
%e 7 .................................. /
%e . /\ /
%e 5 .................... / \ /\/
%e . /\ / \ /\ /
%e 3 .......... / \ / \ / \/
%e 2 ..... /\ / \ /\/ \ /
%e 1 .. /\ / \ /\/ \ / \ /\/
%e . /\/ \/ \/ \/ \/
%e .
%e Note that the j-th largest peak between two valleys at height 0 is also the partition number A000041(j).
%e Written as an irregular triangle in which row k has length 2*A138137(k), the sequence begins:
%e 0,1;
%e 0,1,2,1;
%e 0,1,2,3,2,1;
%e 0,1,2,1,2,3,4,5,4,3,2,1;
%e 0,1,2,3,2,3,4,5,6,7,6,5,4,3,2,1;
%e 0,1,2,1,2,3,4,5,4,3,4,5,6,5,6,7,8,9,10,11,10,9,8,7,6,5,4,3,2,1;
%e 0,1,2,3,2,3,4,5,6,7,6,5,6,7,8,9,8,9,10,11,12,13,14,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1;
%e ...
%Y Column 1 is A000004. Both column 2 and the right border are in A000012. Both columns 3 and 5 are in A007395.
%Y Cf. A000041, A006128, A135010, A138137, A139582, A141285, A182699, A182709, A186412, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610, A229946.
%K nonn,tabf
%O 0,5
%A _Omar E. Pol_, Aug 10 2013