cp's OEIS Frontend

A228110 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.

%I A228110 #45 Apr 01 2017 21:02:02
%S A228110 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 A228110 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 A228110 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 A228110 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 A228110 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 A228110 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 A228110 .
%C A228110 .  j     Diagram 1         Partitions           Diagram 2
%C A228110 .      _ _ _ _ _ _ _                          _ _ _ _ _ _ _
%C A228110 . 15  |_ _ _ _      |      7                  _ _ _ _      |
%C A228110 . 14  |_ _ _ _|_    |      4+3                _ _ _ _|_    |
%C A228110 . 13  |_ _ _    |   |      5+2                _ _ _    |   |
%C A228110 . 12  |_ _ _|_ _|_  |      3+2+2              _ _ _|_ _|_  |
%C A228110 . 11  |_ _ _      | |      6+1                _ _ _      | |
%C A228110 . 10  |_ _ _|_    | |      3+3+1              _ _ _|_    | |
%C A228110 .  9  |_ _    |   | |      4+2+1              _ _    |   | |
%C A228110 .  8  |_ _|_ _|_  | |      2+2+2+1            _ _|_ _|_  | |
%C A228110 .  7  |_ _ _    | | |      5+1+1              _ _ _    | | |
%C A228110 .  6  |_ _ _|_  | | |      3+2+1+1            _ _ _|_  | | |
%C A228110 .  5  |_ _    | | | |      4+1+1+1            _ _    | | | |
%C A228110 .  4  |_ _|_  | | | |      2+2+1+1+1          _ _|_  | | | |
%C A228110 .  3  |_ _  | | | | |      3+1+1+1+1          _ _  | | | | |
%C A228110 .  2  |_  | | | | | |      2+1+1+1+1+1        _  | | | | | |
%C A228110 .  1  |_|_|_|_|_|_|_|      1+1+1+1+1+1+1       | | | | | | |
%C A228110 .
%C A228110 .      1 2 3 4 5 6 7
%C A228110 .
%C A228110 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 A228110 Also the diagram has the property that it can be transformed in a Dyck path (see example).
%C A228110 The sequence gives the height of the infinite Dyck path after n-th step.
%C A228110 The absolute values of the first differences give A000012.
%C A228110 For the height of the peaks and the valleys in the infinite Dyck path see A229946.
%C A228110 Q: Is this infinite Dyck path a fractal?
%H A228110 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa408.jpg">Visualization of regions in a diagram for A006128</a>
%e A228110 Illustration of initial terms (n = 1..59):
%e A228110 .
%e A228110 11 ...........................................................
%e A228110 .                                                            /
%e A228110 .                                                           /
%e A228110 .                                                          /
%e A228110 7 ..................................                      /
%e A228110 .                                  /\                    /
%e A228110 5 ....................            /  \                /\/
%e A228110 .                    /\          /    \          /\  /
%e A228110 3 ..........        /  \        /      \        /  \/
%e A228110 2 .....    /\      /    \    /\/        \      /
%e A228110 1 ..  /\  /  \  /\/      \  /            \  /\/
%e A228110 .  /\/  \/    \/          \/              \/
%e A228110 .
%e A228110 Note that the j-th largest peak between two valleys at height 0 is also the partition number A000041(j).
%e A228110 Written as an irregular triangle in which row k has length 2*A138137(k), the sequence begins:
%e A228110 0,1;
%e A228110 0,1,2,1;
%e A228110 0,1,2,3,2,1;
%e A228110 0,1,2,1,2,3,4,5,4,3,2,1;
%e A228110 0,1,2,3,2,3,4,5,6,7,6,5,4,3,2,1;
%e A228110 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 A228110 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 A228110 ...
%Y A228110 Column 1 is A000004. Both column 2 and the right border are in A000012. Both columns 3 and 5 are in A007395.
%Y A228110 Cf. A000041, A006128, A135010, A138137, A139582, A141285, A182699, A182709, A186412, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610, A229946.
%K A228110 nonn,tabf
%O A228110 0,5
%A A228110 _Omar E. Pol_, Aug 10 2013