On Enumeration of Families of Genus Zero Permutations

Abstract

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