Hypercartes pointées sur le tore: Décompositions et dénombrements

Enumeration of unrooted hypermaps of a given genus

“…The list of 1-admissible orbifolds and the respective numbers Epi o (π 1 (O), C ) were derived in Corollary 4.4. Rooted toroidal maps were enumerated by D. Arquès in [2]. He proved that…”

“…A corresponding case of Problem 2 for g = 0, was settled by Bousquet-Mélou and Schaeffer in terms of planar 2-constellations in 2000 [3]. As concerns genera g ≥ 1, the solution of Problem 3 was obtained by Arqués for g = 1 [2], the other instances of Problem 3 remain unsettled.…”

Character theory and rooted maps in an orientable surface of given genus: face-colored maps

“…denotes its value at any element of ~ where n, 0 f-N. We adopt the convention that the character associated with the empty partition is identically 1 and that X! = 0 if 101 i= Inl· The degree, t , of the irreducible representation indexed by 0 is equal to X~ N] • The rising and falling factorial functions are, respectively,…”

Enumeration of hypermaps of a given genus

“…Let D and D ′ be two lists of integers. The inclusion D ′ ⊆ D means that D ′ is a sublist of D. In this case D − D ′ is the complementary sublist of D ′ in D. For instance, the sublists of D = [1,1,2] are the empty list [ ], [1] (twice), [2], [1,1], [1,2] (twice) and D itself. Their complementary sublists in the same order are D, [1,2] (twice), [1,1], [2], [1] (twice) and [ ].…”

“…This section proposes explicit parametric expressions for the generating functions that count rooted hypermaps of small positive genus. In Section 7.1 we count by number of vertices, hyperedges and faces; the number of darts can be obtained from these parameters by Formula (2). In Section 7.2 we count by number of darts alone.…”