[Paper Review] Asymptotic Enumeration of Labelled Interval Orders

Graham Brightwell, Mitchel T. Keller|arXiv (Cornell University)|Nov 29, 2011

Advanced Combinatorial Mathematics7 references3 citations

TL;DR

This paper establishes an asymptotic formula for the number of labelled interval orders on $n$ elements by leveraging asymptotic analysis of unlabelled rigid interval orders and their connection to Stirling numbers of the second kind. The key result is $\ell_n \sim (n!)^2 \sqrt{n} \left(\frac{6}{\pi^2}\right)^n \frac{12\sqrt{3}}{\pi^{5/2}} \left(1 + O(1/n)\right)$, resolving a long-standing open problem in poset enumeration.

ABSTRACT

Building on work by Zagier, Bousquet-Mélou et al., and Khamis, we give an asymptotic formula for the number of labelled interval orders on an $n$-element set.

Motivation & Objective

• To resolve the asymptotic enumeration of labelled interval orders, a longstanding open problem in poset theory.
• To extend prior work on unlabelled interval orders and rigid unlabelled interval orders to the labelled case.
• To derive precise asymptotic growth rates using generating functions and combinatorial asymptotics.
• To characterize the distribution of duplicated holdings (i.e., automorphism-related symmetries) in random labelled interval orders.

Proposed method

• Use Khamis's generating function for unlabelled rigid interval orders, $R(x) = \sum r_n x^n = \sum_{n\geq 0} \prod_{i=1}^n \left(1 - \frac{1}{(1+x)^i}\right)$, to derive asymptotics for $r_n$.
• Relate $R(x)$ to Bousquet-Mélou et al.'s generating function $I(x)$ via $R(x) = I\left(\frac{x}{1+x}\right)$, enabling coefficient transfer.
• Apply Laplace's method and asymptotic expansion techniques to the coefficient extraction $r_n = \sum_{k=0}^{n-1} (-1)^k \binom{n-1}{k} i_{n-k}$, using the known asymptotic for $i_t$.
• Express the number of labelled interval orders as $\ell_n = \sum_{k=1}^n r_k k! S(n,k)$, where $S(n,k)$ is the Stirling number of the second kind.
• Apply Hsu's asymptotic estimate for $S(n,n-j)$ to analyze the dominant terms in the sum over $j = n-k$.
• Combine the asymptotic expansions of $r_k$ and $S(n,k)$ to derive the final asymptotic formula for $\ell_n$.

Experimental results

Research questions

• RQ1 What is the precise asymptotic growth rate of the number of labelled interval orders on $n$ elements?
• RQ2 How do symmetries (duplicated holdings) distribute in a uniformly random labelled interval order?
• RQ3 What is the relationship between the asymptotics of unlabelled rigid interval orders and the labelled case?
• RQ4 Can the number of pairs of elements with identical upper and lower sets be modeled probabilistically in large random interval orders?
• RQ5 What is the role of the constant $\frac{12\sqrt{3}}{\pi^{5/2}}$ in the leading term of the asymptotic formula?

Key findings

• The number of labelled interval orders on $n$ elements satisfies $\ell_n \sim (n!)^2 \sqrt{n} \left(\frac{6}{\pi^2}\right)^n \frac{12\sqrt{3}}{\pi^{5/2}} \left(1 + O(1/n)\right)$, providing the first precise asymptotic formula.
• The number of unlabelled rigid interval orders satisfies $r_n \sim n! \sqrt{n} \left(\frac{6}{\pi^2}\right)^n \frac{12\sqrt{3}}{\pi^{5/2} e^{\pi^2/12}} \left(1 + O(1/n)\right)$, derived from generating function composition.
• The distribution of the number of pairs of elements with duplicated holdings (i.e., nontrivial automorphisms) is asymptotically Poisson with mean $\pi^2/12$.
• The probability of having a triple of elements with identical upper and lower sets tends to zero as $n \to \infty$, justifying the Poisson approximation.
• The leading constant $E_0 = \frac{12\sqrt{3}}{\pi^{5/2}}$ in the asymptotic for $\ell_n$ arises from $e^{\pi^2/12} D_0$, where $D_0$ is the leading coefficient of $r_n$.
• The method allows for explicit computation of higher-order terms $E_1, E_2, \dots$ in the asymptotic expansion of $\ell_n$ via recursive formulas from the coefficients $C_i$.