Let $S\subset S_{\infty}$ be a set of permutations of $\mathbb{N}$. A real series $\sum_{n\geq0}u_{n}$ will be called $S$-conditionally convergent if it is absolutely divergent and if, for all $\sigma\in S$, $\sum_{n\geq0}u_{\sigma(n)}$ converges.

**Question 1:** Which sets $S$ admit $S$-conditionally convergent series?

By Riemann's rearrangement theorem, $S_{\infty}$ does not admit such a series. Similarly, for any partition $I_{1},\cdots, I_{r}$ of $\mathbb{N}$, $S_{I_{1}}\times\cdots\times S_{I_{r}}$ does not admit such a series, as the same rearrangement argument holds.

On the other hand, any finite set $S=\{\sigma_{1},\cdots,\sigma_{r}\}$ admits an $S$-conditionally convergent series. Indeed, one may find an infinite subset $I$ of $\mathbb{N}$ such that $\sigma_{1}^{-1},\cdots,\sigma_{r}^{-1}$ are monotone upon restriction to $I$, and consider a conditionally convergent series supported in $I$. There is also the notion of a convergence-preserving permutation. Let's say that $\sigma$ satisfies condition $(*)$ if the number of disjoint intervals that make up $\sigma([0,n])$ is bounded with respect to $n$. If all the elements of $S$ satisfy $(*)$ then any conditionally convergent series is $S$-conditionally convergent (see P. Schaefer, "Sum-Preserving Rearrangements of Infinite Series").

One can give a more general sufficient condition: if there is an infinite set $I\subset\mathbb{N}$ such that for every $\sigma\in S$, the permutation $\tilde{\sigma}:\mathbb{N}\cong\sigma^{-1}(I)\rightarrow I\cong\mathbb{N}$ (where the bijections $\cong$ are order-preserving) satisfies $(*)$, then a conditionally convergent series supported in $I$ is $S$-conditionally convergent. Note that under this condition, the sum of the series is also preserved, whereas a set $S$ can admit an $S$-conditionally convergent series whose sum varies depending on the permutation (again by Riemann's rearrangement theorem).

**Question 2:** Can a countable set $S$ admit no $S$-conditionally convergent series? Can one find such a set of the form $\{\sigma^{k};\;k\in\mathbb{Z}\}$?