## General Theory to solve a class of problems

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

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. It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment.