General Theory to solve a class of problems

Arrangements, combinations and permutations, probability, ...
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MuthuVeerappanR
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General Theory to solve a class of problems

Post by MuthuVeerappanR » Mon Dec 07, 2015 11:00 am

A249665 lists the number of permutations p of {1,...,n} such that p(1)=1, p(n)=n, and |p(i)-p(i+1)| is in {1,2,3} for all i from 1 to n-1.

The question can be generalized as 'Find the number of permutations p of {1,...,n} such that p(1)=1, p(n)=n, and |p(i)-p(i+1)| is in some Set A for all i from 1 to n-1'. Additional problems can be like ignoring the 'p(1)=1, p(n)=n' constraints. For example, A174700 and A249665.

The question is How to solve these class of problems in general? What theory does this in a best possible way? Is there any theory to handle this? If yes, can anyone kindly give me the source which shows the solution with examples?

I can't believe there is no general theory to solve this.

Any help is greatly appreciated. Thank You.
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