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A228109
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Height after n-th step of an infinite staircase which is the lower part of a structure whose upper part is the infinite Dyck path of A228110.
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3
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0, -1, 0, -1, 0, -1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 2, 1, 2, 1, 0, -1, 0, -1, 0, 1, 2, 1, 2, 3, 4, 3, 4, 3, 2, 1, 0, -1, 0, -1, 0, 1, 0, 1, 2, 1, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 5, 6, 5, 4, 5, 4, 3, 2, 1, 0, -1
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OFFSET
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0,19
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COMMENTS
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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 = 5, the diagram 1 represents the partitions of 5. The diagram 2 shows separately the boundary segments southwest sides of the first seven regions of the diagram 1, see below:
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j Diagram 1 Diagram 2
_ _ _ _ _
7 |_ _ _ | |_ _ _
6 |_ _ _|_ | |_ _ _ |_
5 |_ _ | | |_ _ |
4 |_ _|_ | | |_ _ |_ |
3 |_ _ | | | |_ _ | |
2 |_ | | | | |_ | | |
1 |_|_|_|_|_| |_ |_ |_ |_ |_
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. 1 2 3 4 5
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a(n) is the height after n-th step of an infinite staircase which is the lower part of a diagram of regions of the set of partitions of all positive integers. The upper part of the diagram is the infinite Dyck path mentioned in A228110. The diagram shows the shape of the successive regions of the set of partitions of all positive integers. The area of the n-th region is A186412(n).
For the height of the peaks and the valleys in the infinite Dyck path see A229946.
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LINKS
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EXAMPLE
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Illustration of initial terms (n = 1..53):
5
4 /
3 /\/\ /
2 / \ /\/
1 /\/\ /\/ \ /\/
0 /\ /\/ \ / \ /\/
-1 \/\/\/\/ \/\/ \/\/ \/\/
-2
The diagram shows the Dyck pack mentioned in A228110 together with the staircase illustrated above. The area of the n-th region is equal to A186412(n).
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7...................................
. /\
5..................... / \ /\
. /\ / \ /\ / /
3........... / \ / /\/\ \ / \/ /
2...... /\ / \ /\/ / \ \ / /\/
1... /\ / \ /\/ /\/\ \ / /\/ \ \ /\/ /\/
0 /\/ \/ /\ \/ /\/ \ \/ / \ \/ /\/
-1 \/\/\/\/ \/\/ \/\/ \/\/
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Region:
. 1 2 3 4 5 6 7 8 9 10
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CROSSREFS
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Cf. A000041, A006128, A135010, A138137, A139582, A141285, A182699, A182709, A186412, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610, A228110, A229946.
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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