On plane permutations

TL;DR

This paper introduces plane permutations, providing a combinatorial proof for a known enumeration result, and explores their applications in studying various permutation distance problems, establishing new connections among them.

Contribution

It generalizes permutations to plane permutations, offers a combinatorial proof for a classical enumeration formula, and links different permutation distance measures using this new framework.

Findings

Derived a combinatorial proof for Zagier and Stanley's enumeration result.

Established connections between transposition, block-interchange, and reversal distances.

Developed recurrences based on a natural transposition action on plane permutations.

Abstract

In this paper we generalize permutations to plane permutations. We employ this framework to derive a combinatorial proof of a result of Zagier and Stanley, that enumerates the number of $n$-cycles $\omega$, for which $\omega(12\cdots n)$ has exactly $k$ cycles. This quantity is $0$, if $n-k$ is odd and $\frac{2C(n+1,k)}{n(n+1)}$, otherwise, where $C(n,k)$ is the unsigned Stirling number of the first kind. The proof is facilitated by a natural transposition action on plane permutations which gives rise to various recurrences. Furthermore we study several distance problems of permutations. It turns out that plane permutations allow to study transposition and block-interchange distance of permutations as well as the reversal distance of signed permutations. Novel connections between these different distance problems are established via plane permutations.