A set-theoretic solution of the Pentagon Equation on a non-empty set S is a map s:S2→S2\documentclass[12pt]{minimal}
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\begin{document}$$s:S^2\rightarrow S^2$$\end{document} such that s23s13s12=s12s23\documentclass[12pt]{minimal}
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\begin{document}$$s_{23}s_{13}s_{12}=s_{12}s_{23}$$\end{document}, where s12=s×id\documentclass[12pt]{minimal}
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\begin{document}$$s_{12}=s\times {{{\,\mathrm{id}\,}}}$$\end{document}, s23=id×s\documentclass[12pt]{minimal}
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\begin{document}$$s_{23}={{{\,\mathrm{id}\,}}}\times s$$\end{document} and s13=(τ×id)(id×s)(τ×id)\documentclass[12pt]{minimal}
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\begin{document}$$s_{13}=(\tau \times {{{\,\mathrm{id}\,}}})({{{\,\mathrm{id}\,}}}\times s)(\tau \times {{{\,\mathrm{id}\,}}})$$\end{document} are mappings from S3\documentclass[12pt]{minimal}
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\begin{document}$$S^3$$\end{document} to itself and τ:S2→S2\documentclass[12pt]{minimal}
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\begin{document}$$\tau :S^2\rightarrow S^2$$\end{document} is the flip map, i.e., τ(x,y)=(y,x)\documentclass[12pt]{minimal}
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\begin{document}$$\tau (x,y) =(y,x)$$\end{document}. We give a description of all involutive solutions, i.e., s2=id\documentclass[12pt]{minimal}
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\begin{document}$$s^2={{\,\mathrm{id}\,}}$$\end{document}. It is shown that such solutions are determined by a factorization of S as direct product X×A×G\documentclass[12pt]{minimal}
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\begin{document}$$X\times A \times G$$\end{document} and a map σ:A→Sym(X)\documentclass[12pt]{minimal}
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\begin{document}$$\sigma :A\rightarrow {{\,\mathrm{Sym}\,}}(X)$$\end{document}, 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 σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document} is irrelevant. In particular, if S is finite of cardinality 2n(2m+1)\documentclass[12pt]{minimal}
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\begin{document}$$2^n(2m+1)$$\end{document} for some n,m⩾0\documentclass[12pt]{minimal}
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\begin{document}$$n,m\geqslant 0$$\end{document} then, on S, there are precisely n+22\documentclass[12pt]{minimal}
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\begin{document}$$\left( {\begin{array}{c}n+2\\ 2\end{array}}\right) $$\end{document} non-isomorphic solutions of the Pentagon Equation.
