|
| |
|
|
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.
|
|
3
|
|
|
|
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, 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, 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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
COMMENTS
|
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:
.
. 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 | | | | | | |
.
. 1 2 3 4 5 6 7
.
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 sequence gives the height of the infinite Dyck path after n-th step.
The absolute values of the first differences give A000012.
For the height of the peaks and the valleys in the infinite Dyck path see A229946.
Q: Is this infinite Dyck path a fractal?
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
Illustration of initial terms (n = 1..59):
.
11 ...........................................................
. /
. /
. /
7 .................................. /
. /\ /
5 .................... / \ /\/
. /\ / \ /\ /
3 .......... / \ / \ / \/
2 ..... /\ / \ /\/ \ /
1 .. /\ / \ /\/ \ / \ /\/
. /\/ \/ \/ \/ \/
.
Note that the j-th largest peak between two valleys at height 0 is also the partition number A000041(j).
Written as an irregular triangle in which row k has length 2*A138137(k), the sequence begins:
0,1;
0,1,2,1;
0,1,2,3,2,1;
0,1,2,1,2,3,4,5,4,3,2,1;
0,1,2,3,2,3,4,5,6,7,6,5,4,3,2,1;
0,1,2,1,2,3,4,5,4,3,4,5,6,5,6,7,8,9,10,11,10,9,8,7,6,5,4,3,2,1;
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;
...
|
|
|
CROSSREFS
|
Column 1 is A000004. Both column 2 and the right border are in A000012. Both columns 3 and 5 are in A007395.
Cf. A000041, A006128, A135010, A138137, A139582, A141285, A182699, A182709, A186412, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610, A229946.
|
|
|
KEYWORD
|
nonn,tabf
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
