Plane Permutations and Applications to a Result of Zagier--Stanley and Distances of Permutations

Abstract

In this paper, we introduce plane permutations, i.e., pairs $\mathfrak{p}=(s,\pi)$, where $s$ is an $n$-cycle and $\pi$ is an arbitrary permutation, represented as a two-row array. Accordingly, a plane permutation gives rise to three distinct permutations: the permutation induced by the upper horizontal ($s$), the vertical ($\pi$), and the diagonal ($D_{\mathfrak{p}}$) of the array. The latter can also be viewed as the three permutations of a hypermap. In particular, a map corresponds to a plane permutation, in which the diagonal is a fixed-point free involution. We study the transposition action on plane permutations obtained by permuting their diagonal blocks. We establish basic properties of plane permutations and analyze transpositions and exceedances, deriving various enumerative results. In particular we prove a recurrence for the number of plane permutations having a fixed diagonal and $k$ cycles in the vertical, generalizing a recursion of Chapuy for maps filtered by the genus. As applications of ...