  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.
