The Riemann Rearrangement Theorem is very well-known. It states that a conditionally convergent series may have its terms permuted to give any real sum as well as $\infty$ and $-\infty$ and we can also give divergence. However, certain rearrangements will always leave a sum fixed. For example, changing the position of only finitely many terms works, so does partitioning $\mathbb{N}$ into contiguous blocks of bounded length, and permuting each of those separately. Is there a characterisation of these permutations of $\mathbb{N}$?
Formally, which permutations $p:\mathbb{N}\to\mathbb{N}$ are such that for all real-valued sequences ${(s_i)}_{i=0}^\infty$ such that the series $\sum_{i=0}^\infty s_i$ is convergent, we have $\sum_{i=0}^\infty s_i=\sum_{i=0}^\infty s_{p(i)}$. What if we allow infinite sums?
