[Paper Review] Another combinatorial proof of a result of Zagier and Stanley

Ricky X. F. Chen, Christian M. Reidys|arXiv (Cornell University)|Feb 26, 2015

Genome Rearrangement Algorithms34 references3 citations

TL;DR

This paper presents a new combinatorial proof for a result by Zagier and Stanley on the number of $n$-cycles $\omega$ such that $\omega(12\cdots n)$ has exactly $k$ cycles, showing it is zero when $n-k$ 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 introduces plane permutations and uses a transposition action to analyze exceedances, while also refining a recurrence for one-face hypermaps via a reflection principle.

ABSTRACT

In this paper, we present another combinatorial proof for the result of Zagier and Stanley, that the number of $n$-cycles $\omega$, for which $\omega(12\cdots n)$ has exactly $k$ cycles 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. To this end, we generalize permutations to plane permutations and study exceedances via a natural transposition action on plane permutations. Furthermore, based on a reflection principle argument, we give a refinement of a recurrence satisfied by the numbers counting one-face hypermaps which was recently obtained by Chapuy by counting bipartite unicellular maps.

Motivation & Objective

• To provide an alternative combinatorial proof for a known result on cycle counts in permutations.
• To generalize permutations to plane permutations to facilitate the analysis of exceedances.
• To refine a recurrence for one-face hypermaps using a reflection principle argument.
• To establish a connection between transposition actions on plane permutations and cycle structure in products of permutations.

Proposed method

• Introduce the concept of plane permutations as a generalization of standard permutations.
• Define a transposition action on plane permutations to study exceedances and their distribution.
• Apply a reflection principle to derive a refined recurrence for the number of one-face hypermaps.
• Use generating functions and combinatorial identities to relate plane permutations to Stirling numbers of the first kind.
• Leverage the structure of the product $\omega(12\cdots n)$ to analyze cycle counts via combinatorial symmetries.
• Establish a bijection-like argument through transposition dynamics to prove the vanishing condition when $n-k$ is odd.

Experimental results

Research questions

• RQ1 What combinatorial structure underlies the vanishing of cycle counts when $n-k$ is odd in the product $\omega(12\cdots n)$?
• RQ2 How can plane permutations be used to model and analyze exceedances in permutation products?
• RQ3 Can a reflection principle be applied to refine the recurrence for one-face hypermaps recently derived by Chapuy?
• RQ4 What is the precise combinatorial interpretation of the factor $\frac{2C(n+1,k)}{n(n+1)}$ in the cycle count formula?
• RQ5 How does the transposition action on plane permutations encode information about cycle structure in $\omega(12\cdots n)$?

Key findings

• The number of $n$-cycles $\omega$ for which $\omega(12\cdots n)$ has exactly $k$ cycles is zero when $n-k$ is odd.
• When $n-k$ is even, the number of such $\omega$ is $\frac{2C(n+1,k)}{n(n+1)}$, where $C(n+1,k)$ is the unsigned Stirling number of the first kind.
• The introduction of plane permutations provides a new framework for analyzing exceedances through a transposition action.
• A refined recurrence for one-face hypermaps is derived using a reflection principle, extending a result by Chapuy.
• The transposition action on plane permutations reveals structural symmetries that explain the vanishing condition for odd $n-k$.
• The proof establishes a direct combinatorial link between plane permutations and the cycle index of the product $\omega(12\cdots n)$.