Consider the array $\{x_{i,n}:1\leq i\leq n, n\in\mathbb{N}\}$ and the set of consecutive indices $A_n$ whose cardinality is $d_n$, where $d_n<n$ is an increasing positive sequence (slower than $n$). Say that $A_n=\{c_n,\dotsc,c_n+d_n-1\}$ with $c_n$ being a positive sequence such that $c_n+d_n-1\leq n$. Then, for a given $n$, all of the following $n$-tuples $$(0,0,c_n,\dotsc,c_n+d_n-1,0,0),(0,0,0,c_n,\dotsc,c_n+d_n-1,0),(0,0,0,0,c_n,\dotsc,c_n+d_n-1,(c_n,\dotsc,c_n+d_n-1,0,0,0,0)$$ are associated with same sum $\sum_{i\in A_n}x_{i,n}$. So I think of this situation as if there were an equivalence class $S_n$ to represent all these n-tuples. How could I define such equivalence relation of "interchanging zeros" n-tuples.
With the sets $S_n$ stablished, we see that the sub-array $\{x_{i,n}:i\in A_n, n\in\mathbb{N}\}$ and any other $\{x_{i,n}:i\in I_n, n\in\mathbb{N}\}$, give the same sum $$\sum_{i\in A_n}x_{i,n}=\sum_{i\in I_n}x_{i,n}, \forall I_n\in S_n$$ in spite of they are unequal arrays.
I appreciate any help to elaborate this device, properly.