A combinatorial relationship between Eulerian maps and hypermaps in orientable surfaces (Q1296755)

This paper studies maps, graphs cellularly embedded on orientable surfaces. The genus series is the generating function \(M(u,x,y,z)\) where the coefficient on \(u^g x^i y^j z^k\) is the number \(m_{g,i,j,k}\) of rooted maps of genus \(g\) with \(i\) vertices, \(j\) faces, and \(k\) edges (any one of these parameters is superfluous, but all are included for symmetry). Using character theory the authors had previously determined an interesting relationship between the genus series for rooted maps and the genus series for rooted quadrangulations. The authors generalize this relationship to a larger set of maps where the relationship can be described. They examine properties of a putative bijection.