Partial-dual Euler-genus distributions for bouquets with small Euler genus

For an arbitrary ribbon graph G, the partial-dual Euler-genus polynomial of G is a generating function that enumerates partial duals by Euler genus. When orientable graph, genus Gross, Mansour, and Tucker inaugurated these distribution problems in 2020. A bouquet one-vertex graph. Given its same as some bouquet; this motivates our focus on bouquets. We obtain polynomials for all bouquets with at most two.