# Linear extension numbers of $n$-element posets

## TL;DR

This paper investigates the possible counts of linear extensions of n-element posets, revealing that these counts are densely distributed near 1 and sparse near n! with connections to number theory.

## Contribution

It characterizes the set of linear extension numbers for n-element posets, showing their distribution and density within the interval [1, n!], and introduces number-theoretic methods involving the Stern-Brocot tree.

## Key findings

- All integers up to exp(c n / log n) are realizable as linear extension counts.
- The number of such counts in the interval ((n-1)!, n!] is less than (n-3)!.
- The proof involves novel number-theoretic results related to the Stern-Brocot tree.

## Abstract

We address the following natural but hitherto unstudied question: what are the possible linear extension numbers of an $n$-element poset? Let $\mathbf{LE}(n)$ denote the set of all positive integers that arise as the number of linear extensions of some $n$-element poset. We show that $\mathbf{LE}(n)$ skews towards the "small" end of the interval $[1,n!]$. More specifically, $\mathbf{LE}(n)$ contains all of the positive integers up to $\exp\left(c\frac{n}{\log n}\right)$ for some absolute constant $c$, and $|\mathbf{LE}(n) \cap ((n-1)!,n!]|<(n-3)!$. The proof of the former statement involves some intermediate number-theoretic results about the Stern-Brocot tree that are of independent interest.