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The axiomatizations provide an encoding of rooted maps as a permutation such that the face permutation is obtained as a product of two permutations. For maps in orientable surfaces it is therefore natural to consider the class algebra of the symmetric group, whose conjugacy classes Cλ are indexed by partitions of n as a natural index. These partitions then correspond to vertex and face partitions. Orthogonal idempotents can be constructed that span the class algebra. These are linear combinations of the Cλ ’s, where Cλ is a formal sum of the elements of Cλ .

Then τ is the permutation whose disjoint cycles are associated in pairs with each vertex v, and have the form (a1, . . , ak ). The degree of v is k. The permutation τ is called the vertex permutation or the rotation system for the map in an orientable surface. It is convenient to regard the two ends of an edge as belonging, notionally, to two different halves of the same edges. These two halves are called half-edges. 1. Orientable surfaces ¡ 4 10 5 1 8 6 7 9 2 31 3 illustrates how the faces and face partition are encoded by these actions in the specific case of the labelling of m1·457 .

Now τ has cycle type [34], and dividing the multiplicities by 2 (the cycles appear in pairs) gives [32 ], so there are two vertices of degree 3. Similarly the cycle type of ϕ is [62 ], and dividing the multiplicities by 2 gives [6], so there is only one face and this has degree 6. In staying on the same side of the boundary in a tour of a face (1 5 6 4 11 12) is encountered as the list of labels on the tails of edges, and (2 8 7 3 10 9) as the heads of edges. These two cycles have the canonically defined senses (1, ρ(1)) and (2, ρ(2)), respectively, and these are opposite senses.