On Enumeration of Families of Genus Zero Permutations

The genus of a permutation σ of length n is the nonnegative integer g_σ given by n+ 1 - 2 g_σ= cyc(σ) + cyc(σ^{- 1}ζ_n) , where cyc(σ) is the number of cycles of σ and ζ_n is the cyclic permutation (1 , 2 , … , n). On the basis of a connection between genus zero permutations and noncrossing partitions, we enumerate the genus zero permutations with various restrictions, including André permutations, simsun permutations, and smooth permutations. Moreover, we present refined sign-balance results on genus zero permutations and their analogues restricted to connected permutations.