Updown numbers and the initial monomials of the slope variety

Details Updown numbers and the initial monomials of the slope variety Updown numbers and the initial monomials of the slope variety

2009-05-28

arxiv.org/abs/0905.4751v1

Electronic J. Combin. 16, no. 1 (2009), R82

Mathematics

Combinatorics

7 pages; 1 figure

Scientific paper

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.

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