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## Exercise V

Posted 06/01/2016
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This exercise is about the number of linear partitions of 2-by-n grid points.

We consider a set of lattice points, \{1,2,\cdots,n\}\times\{1,2\}. For example, the set of 2-by-4 gird points are pictured as follows.

We want to partition these 2n points into two nonempty parts by a straight line. An example of two different linear partitions of 2-by-4 grid points is pictured as follows.

Note that the area/size of grid points are not taken into consideration.

Suppose a(n) is the number of linear partitions of such 2\times n grid points.

For example, a(1)=1 and a(2)=6.

Find a(n).