Updown numbers and the initial monomials of the slope variety

TL;DR

This paper investigates the algebraic relations among slopes of lines determined by points in a plane, revealing a combinatorial structure linked to permutation pattern avoidance and Euler numbers.

Contribution

It establishes a bijective correspondence between initial ideal generators and pattern-avoiding permutations counted by Euler numbers, providing a new combinatorial formula.

Findings

Initial ideals are generated by monomials linked to pattern-avoiding permutations.

Number of generators in each degree is given by Euler numbers.

Provides a combinatorial enumeration of algebraic relations in slope varieties.

Abstract

Let $I_n$ be the ideal of all algebraic relations on the slopes of the $\binom{n}{2}$ lines formed by placing $n$ points in a plane and connecting each pair of points with a line. Under each of two natural term orders, the initial ideal of $I_n$ is generated by monomials corresponding to permutations satisfying a certain pattern-avoidance condition. We show bijectively that these permutations are enumerated by the updown (or Euler) numbers, thereby obtaining a formula for the number of generators of the initial ideal of $I_n$ in each degree.