Set-theoretic solutions of the Pentagon Equation

A set-theoretic solution of the Pentagon Equation on a non-empty set $S$ is a map $s\colon S^2\to S^2$ such that $s_{23}s_{13}s_{12}=s_{12}s_{23}$, where $s_{12}=s\times\mathrm{id}$, $s_{23}=\mathrm{id}\times s$ and $s_{13}=(\tau\times\mathrm{id})(\mathrm{id}\times s)(\tau\times\mathrm{id})$ are mappings from $S^3$ to itself and $\tau\colon S^2\to S^2$ is the flip map, i.e., $\tau (x,y) =(y,x)$. We give a description of all involutive solutions, i.e., $s^2=\mathrm{id}$. It is shown that such solutions are determined by a factorization of $S$ as direct product $X\times A \times G$ and a map $\sigma\colon A\to\mathrm{Sym}(X)$, where $X$ is a non-empty set and $A,G$ are elementary abelian $2$-groups. Isomorphic solutions are determined by the cardinalities of $A$, $G$ and $X$, i.e., the map $\sigma$ is irrelevant. In particular, if $S$ is finite of cardinality $2^n(2m+1)$ for some $n,m\geq 0$ then, on $S$, there are precisely $\binom{n+2}{2}$ non-isomorphic solutions of the Pentagon Equation.