Tilings and matroids on regular subdivisions of a triangle

TL;DR

This paper explores the structure of a specific family of matroids related to lattice points in a triangle, connecting their properties with tilings into geometric shapes like triangles, rhombi, and trapezoids.

Contribution

It characterizes the independent sets, circuits, and flats of the matroids  in relation to geometric tilings of the triangle.

Findings

Characterization of independent sets of .

Connection between rank function and tilings.

Geometric description of flats of .

Abstract

In this paper we investigate a family of matroids introduced by Ardila and Billey to study one-dimensional intersections of complete flag arrangements of $\mathbb{C}^n$. The set of lattice points $P_n$ inside the equilateral triangle $S_n$ obtained by intersecting the nonnegative cone of $\mathbb{R}^3$ with the affine hyperplane $x_1 + x_2 + x_3 = n-1$ is the ground set of a matroid $\mathcal{T}_n$ whose independent sets are the subsets $S$ of $P_n$ satisfying that $|S \cap P| \le k$ for each translation $P$ of the set $P_k$. Here we study the structure of the matroids $\mathcal{T}_n$ in connection with tilings of $S_n$ into unit triangles, rhombi, and trapezoids. First, we characterize the independent sets of $\mathcal{T}_n$, extending a characterization of the bases of $\mathcal{T}_n$ already given by Ardila and Billey. Then we explore the connection between the rank function of $\mathcal{T}_n$ and the tilings of $S_n$ into unit triangles and rhombi. Then we provide a tiling characterization of the circuits of $\mathcal{T}_n$. We conclude with a geometric characterization of the flats of $\mathcal{T}_n$.