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arXiv · 0905.4751

Updown numbers and the initial monomials of the slope variety

Jeremy L. Martin·Jennifer D. Wagner
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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.

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Jeremy L. Martin, Jennifer D. Wagner. 2009-05-28. Updown numbers and the initial monomials of the slope variety. https://arxiv.org/abs/0905.4751

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