[Paper Review] Enumeration of chord diagrams

Andrei Khruzin|ArXiv.org|Aug 28, 2000

Advanced Combinatorial Mathematics5 references8 citations

TL;DR

This paper enumerates chord diagrams of order $n$ under the action of cyclic ($C_{2n}$) and dihedral ($D_{2n}$) groups, deriving exact formulas and asymptotic estimates for the number of non-equivalent diagrams. It uses group action and cycle index techniques to compute orbit counts, showing that asymptotically, each equivalence class contains $2n$ diagrams under $C_{2n}$ and $4n$ under $D_{2n}$, with the counts closely approximating $(2n-1)!!/(2n)$ and $(2n-1)!!/(4n)$, respectively.

ABSTRACT

We determine the number of nonequivalent chord diagrams of order $n$ under the action of two groups, $C_{2n}$, a cyclic group of order $2n$, and $D_{2n}$, a dihedral group of order $4n$. Asymptotic formulas are also established.

Motivation & Objective

• To determine the number of non-equivalent chord diagrams of order $n$ under the action of the cyclic group $C_{2n}$.
• To extend the enumeration to the dihedral group $D_{2n}$, which includes reflections.
• To derive exact formulas and asymptotic estimates for the number of such non-equivalent diagrams.
• To analyze the stabilizer structure and orbit sizes under group actions, showing that most diagrams have trivial stabilizers asymptotically.

Proposed method

• The paper applies the cycle index method from group theory to count orbits of 1-factors (perfect matchings) under group actions.
• It uses the wreath product $S_n \wr S_2$ to model the symmetry of chord pairings and the group $G$ (either $C_{2n}$ or $D_{2n}$) to model symmetries of the circle.
• A key formula (1) computes the number of orbits as a double sum over group elements and matching symmetries, using cycle types and falling factorials.
• For $C_{2n}$, the formula simplifies to $c_n = \frac{1}{2n} \sum_{i|2n} \varphi(i) \nu_n(i)$, where $\nu_n(i)$ depends on parity of $i$.
• For $D_{2n}$, the formula is derived by adding contributions from reflections, leading to $d_n = \frac{1}{2}\left(c_n + \frac{1}{2}(\kappa_{n-1} + \kappa_n)\right)$, with $\kappa_n$ a sum over matchings with fixed points.
• Asymptotic analysis uses Stirling's formula and bounds on binomial and double factorial terms to show that $c_n \sim (2n-1)!! / (2n)$ and $d_n \sim (2n-1)!! / (4n)$.

Experimental results

Research questions

• RQ1 What is the exact number of non-equivalent chord diagrams of order $n$ under the action of the cyclic group $C_{2n}$?
• RQ2 How does the number of non-equivalent diagrams change when the dihedral group $D_{2n}$ is used instead of $C_{2n}$?
• RQ3 What is the asymptotic behavior of the number of non-equivalent $n$-diagrams as $n \to \infty$?
• RQ4 How large are the equivalence classes (orbits) under $C_{2n}$ and $D_{2n}$, and what fraction of diagrams have nontrivial stabilizers?

Key findings

• The number of non-equivalent $n$-diagrams under $C_{2n}$ is given by $c_n = \frac{1}{2n} \sum_{i|2n} \varphi(i) \nu_n(i)$, with $\nu_n(i)$ defined piecewise for odd and even $i$.
• For $D_{2n}$, the number is $d_n = \frac{1}{2}\left(c_n + \frac{1}{2}(\kappa_{n-1} + \kappa_n)\right)$, where $\kappa_n = \sum_{k=0}^{\lfloor n/2 \rfloor} \frac{n!}{k! (n-2k)!}$.
• Asymptotically, $c_n \sim \frac{(2n-1)!!}{2n}$, meaning each orbit contains approximately $2n$ diagrams.
• Similarly, $d_n \sim \frac{(2n-1)!!}{4n}$, so each orbit under $D_{2n}$ contains about $4n$ diagrams.
• The fraction of diagrams with nontrivial stabilizers in $C_{2n}$ tends to zero as $n \to \infty$, confirming that most diagrams are asymmetric.
• Numerical tables confirm the asymptotic estimates: for $n=10$, $c_n \approx 32.7$ million and $\underline{c}_n \approx 32.7$ million, showing strong agreement.