Plane permutations and applications to a result of Zagier-Stanley and distances of permutations

TL;DR

This paper introduces plane permutations, a new combinatorial framework involving pairs of permutations, and applies it to derive results on permutation distances, map enumeration, and a classic problem by Zagier and Stanley.

Contribution

The paper develops the theory of plane permutations, generalizes map enumeration, and provides a combinatorial proof of Zagier-Stanley's result, connecting permutation distances and hypermap structures.

Findings

Established properties of plane permutations and their transpositions

Derived a recurrence for plane permutations with fixed diagonals and cycles

Connected permutation distances with hypermap and map enumeration

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 study transpositions and exceedances and derive 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 Chapuy's recursion for maps filtered by the genus. As applications of this framework, we present a combinatorial proof of a result of Zagier and Stanley, on the number of $n$-cycles $\omega$, for which the product $\omega(1~2~\cdots ~n)$ has exactly $k$ cycles. Furthermore, we integrate studies on the transposition and block-interchange distance of permutations as well as the reversal distance of signed permutations. Plane permutations allow us to generalize and recover various lower bounds for transposition and block-interchange distances and to connect reversals with block-interchanges.