2018/02/15 by Felix Gotti, Gotti, Felix, Harold Polo +1
Engineering · Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1802.05633
openalex publication_date 2018/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we investigate a family of matroids introduced by Ardila and Billey to study one-dimensional intersections of complete flag arrangements of ℂn. The set of lattice points Pn inside the equilateral triangle Sn obtained by intersecting the nonnegative cone of ℝ3 with the affine hyperplane x1 + x2 + x3 = n-1 is the ground set of a matroid Tn whose independent sets are the subsets S of Pn satisfying that |S ∩ P| ≤ k for each translation P of the set Pk. Here we study the structure of the matroids Tn in connection with tilings of Sn into unit triangles, rhombi, and trapezoids. First, we characterize the independent sets of Tn, extending a characterization of the bases of Tn already given by Ardila and Billey. Then we explore the connection between the rank function of Tn and the tilings of Sn into unit triangles and rhombi. Then we provide a tiling characterization of the circuits of Tn. We conclude with a geometric characterization of the flats of Tn.