A face polynomial invariant for embedded graphs

We introduce the face polynomial of a ribbon graph, a generating function for face numbers under the ribbon group action. It addresses the face-count case of the Ellis–Monaghan–Moffatt question on twisted duality. We derive deletion–contraction recursions and prove that this polynomial coincides with the vertex polynomial for $\langleδ\rangle$, $\langleδτ\rangle$, and $\langleδ,τ\rangle$. We characterize monomial cases and obtain formulas and recursions for Euler-genus partial-twuality polynomials. For orientable graphs, we show that the partial-dual face polynomial has both parities whenever an edge exists, so the partial-dual Euler-genus polynomial has only even exponents.