This possibly easy question is related to  [this one.](https://mathoverflow.net/questions/102933/reconstructing-a-word)  Let $s\_1,...,s\_n$ be a sequence of natural numbers (some of them may be equal to 0). Consider the following sequence of multisets of 2-vectors (each vector is counted with its multiplicity) of natural numbers $A\_1=\{ (s\_1,s\_2),...(s\_n,s\_1)\}$, $A\_2=\{(s\_1+s\_2,s\_3),...,(s\_n+s\_1,s\_2)\}$, ...,$A\_n=\{(s\_1+...+s\_n,0)\}$. 
Note that each multiset is closed under taking cyclic shift on the set of indices $1,2,...,n$.


 **Question** Does this sequence of multisets determine the sequence $s\_1,...,s\_n$ up to a cyclic shift?


In particular, is the [Prouhet-Morse-Thue](http://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence) sequence reconstructible (up to a cyclic shift)?


 **Example**  If we encode every pair $(i,j)$ by a monomial $a^ib^j$ in commuting variables $a,b$, and interpret the multiplicity as a coefficient, we encode every multiset as a polynomial in $a,b$ over $\mathbb{Z}$. Here are the first 7 polynomials corresponding to the Prouhet-Morse-Thue word of length $8$ p\_8="10010110":


$$3a+1+3b+ab\\\
3a+b+3ab+{a}^{2}\\\
2ab+2a+2{a}^{2}b+2{a}^{2}\\\
2{a}^{2}+2\,ab+2{a}^{2}b+2\,{a}^{3}\\\
3{a}^{2}b+3{a}^{3}+{a}^{3}b+{a}^{2}\\\
3{a}^{3}b+3{a}^{3}+{a}^{4}+{a}^{2}b\\\\
4{a}^{4}+4{a}^{3}b$$


Does this sequence of polynomials determine the word $p\_8$ up to a cyclic shift?  **Update**  Answer for $p\_8$ is "yes" (computed using Maple).



A combinatorial question
Unless I made mistake a counterexample is formed by the binary $m$-sequence of length 7 $s=(1,0,0,1,0,1,1)$ and its reversal $\tilde{s}=(1,1,0,1,0,0,1)$ that is not a cyclic shift of $s$. Both lead to the sequence of generating functions $1+2a+2b+2ab$, $2a+b+a^2+2ab+a^2b$, $a+a^2+2ab+a^3+2a^2b$, $a^2+ab+2a^3+2a^2b+a^3b$, $2a^3+2a^2b+a^4+2a^3b$ and $3a^4+4a^3b$.


The $m$-sequences are examples of de Bruijn -sequences. That is binary sequences of length $2^n-1$ such that every sequence of $n$ bits (with the exception of $n$ zeros) occurs exactly once in the cycle. This is, of course, then a natural source for an eventual counterexample as the condition is automatically satisfied for $A\_j, j\lt n.$ Length 7 is the shortest, where not all de Bruijn sequences are cyclic shifts of each other. 


An $m$-sequence is generated by a linear feedback shift register that has a primitive polynomial of degree $n$ from the ring $\mathbb{F}\_2[x]$ as a feedback polynomial. If you decimate an $m$-sequence with a decimation exponent $d$, $\gcd(d, 2^n-1)=1,$ by cyclically taking every $d^{th}$ member, you get another $m$-sequence. Therefore for larger $n$ the number of non-cyclically equivalent $m$-sequences increases. For example, the sequence $s$
is generated by the feedback polynomial $x^3+x^2+1$, or equivalently by the given first 3 bits and the recurrence relation $$s\_n=s\_{n-3}+s\_{n-2}\pmod2$$
with subscript arithmetic done modulo seven. The reversed sequence $\tilde{s}$ is similarly generated by the reciprocal polynomial $x^3+x+1$.


I dare not guess yet, whether all $m$-sequences of a given length give rise to the same sequence of multisets.


