My question is the following: Let $C:= \{(f_1,f_2,f_3,..): \text{each } f_i \text{ is a non-negative integer and only finitely many nonzero}\}$. I am looking for an involution $\phi: C \xrightarrow{} C$ such that
$\phi \circ \phi = \text{identity on C}$
Moreover, the sum $\sum_{i=1}^{\infty}f_i \cdot i$ is invariant under $\phi$. In other words, let $f = (f_1,f_2,f_3,\cdots) \in C$ and $\phi(f) = (g_1,g_2,g_3,\cdots) \in C$. Then, $\sum_{i=1}^{\infty} f_i \cdot i = \sum_{i=1}^{\infty} g_i \cdot i$.
We want the set of fixed points to be as small as possible, i.e we want the set $\{f \in C: \phi(f)=f\}$ to be small.
Obviously, I am looking for something which is not the identity function:)
Motivation: I am not sure whether the motivation is related to the answer of the question or not, let me give it anyway. There is a well-known involution in integer partitions, namely the conjugation. It interchanges rows and columns of the Ferrer diagram of a partition. An integer partition $\lambda$ can be represented as a sequence $(f_1,f_2,f_3, \cdots)$ in $C$, where $C$ is defined as above. Intuitively, $f_i$ is the number of times, the part $i$ appears in $\lambda$.
For example, $5 + 5 + 3 + 2 + 2 + 2 + 1$ can be represented as $(1 , 3 , 1 , 0 , 2, 0 , 0 , \cdots)$.
It seems to me, the conjugation operation is not easily representable using the sequence representation of the partition. Therefore, I am looking for an involution which is easily representable("more natural") using sequence representation. The requirement for the invariance of $\sum_{i=1}^{\infty}f_i \cdot i$ comes from integer partition motivation.
PS: If you have other ideas, which does not satisfy the requirements but the function $\phi$ is natural to consider in some other areas of mathematics, I want to learn them as well.
Thanks in advance.