Updown Numbers and the Initial Monomials of the Slope Variety

Abstract

Let $I_n$ be the ideal of all algebraic relations on the slopes of the ${n\choose2}$ 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 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.