Genus distributions for bouquets of circles (Q1263599)

Two imbeddings f: \(G\to S\) and g: \(G\to S\) of a graph G into a surface S are regarded as being different if there is no orientation-preserving homeomorphism h: \(S\to S\) such that \(hf=g\). The genus distribution of G is the sequence \(\{g_ m(G)\}\), where \(g_ m(G)\) is the number of different imbeddings of G into the closed orientable 2-manifold \(S_ m\) of genus m. (Thus it is labeled imbeddings which are being counted.) A bouquet of circles \(B_ n\) consists of one vertex and n loops at that vertex, \(n\in N\). A counting formula of \textit{D. M. Jackson} concerning the cycle structure of permutations [Counting cycles in permutations by group characters, with an application to a topological problems, Trans. Am. Math. Soc. 299, 785-801 (1987; Zbl 0655.05005)] is used to derive \(\{g_ m(B_ n)\}\), for each n. The authors show that all of these genus distributions for bouquets are strongly unimodal.