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A367783
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Number of subsets of the integer lattice Z^2 of cardinality n such that there is no monotone lattice path which splits the set in half, up to shifts.
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1
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0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 0, 4, 0, 6, 0, 6, 0, 11, 0, 12, 8, 7, 0, 6, 0, 8, 0, 18, 0, 32, 0, 20, 0, 29, 0, 42, 8, 67, 16, 30, 0, 13, 0, 22, 0, 32, 0, 42, 0, 64, 0, 50, 0, 64
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OFFSET
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1,8
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COMMENTS
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A monotone path is a lattice path consisting of east and north unit steps or a path consisting of east and south unit steps. When counting, points lying on the path itself are discarded.
a(n) > 0 for even n >= 12.
a(n) > 0 for odd n with natural density 1 (among odd numbers).
For odd n, a(n) is divisible by 8.
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LINKS
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Giedrius Alkauskas, Problem 11484, Problems and solutions, Amer. Math. Monthly, 117 (2) February (2010), p. 182.
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EXAMPLE
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For n = 4 a(4) = 1 way to place 4 points is as follows:
.xx.
.xx.
For n = 8 a(8) = 2 ways to place 8 points are as follows:
..x.
.xxx
xxx.
.x..
(and its reflection with respect to a vertical axis).
For n = 18 a(18) = 4 ways to place 18 points are as follows:
...x..
..xxx.
.xxxxx
xxxxx.
.xxx..
..x...
(and its reflection with respect to a vertical axis), and
.....x....
......x...
.......x..
....x...x.
...xxx...x
x...xxx...
.x...x....
..x.......
...x......
....x.....
(and its reflection with respect to a vertical axis).
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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