My interest in combinatorially motivated computational problems led me to search for simple problems that turn out to be computationally hard. In this pursuit, I came up with a problem which I hope is NP-complete. I searched the literature without finding an equivalent or close problem.

Originally motivated by this post, Can you identify the sum of two permutations in polynomial time? , and my interest in computational properties of permutations.

A differences sequence $a_1, a_2, \ldots a_n$ of a permutation $\pi$ of numbers $1, 2, \ldots n+1$ is formed by finding the difference between every two adjacent numbers in the permutation $\pi$. In other words, $a_i= |\pi(i+1)-\pi(i)|$ for $1 \le i \le n$

For example, sequence $1, 1, 3$ is the differences sequence of permutation $2 3 4 1$. While, sequences $2, 2, 3$ and $ 3, 1, 2$ are not the differences sequence of any permutation of numbers $1, 2, 3, 4$.

Is there an efficient algorithm to determine whether a given sequence is the differences sequence for some permutation $\pi$, or is it NP-hard?

We get computationally equivalent problem if we formulate the problem using circular permutations.

Cross posted on TCS SE without an answer. Efficient algorithm for existence of permutation with differences sequence?

**EDIT** Marzio's nice NP-completeness proof has been published in the Electronic Journal of Combinatorics.