This possibly easy question is related to this one. 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 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).