Equal Entries in Totally Positive Matrices

2013/09/17 by Miriam Farber, Farber, Miriam, Mitchell Faulk +5
Computer Science · Engineering · Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1309.4186

openalex publication_date 2013/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the maximal number of equal entries in a totally positive (resp. totally nonsingular) n\textrm-by-n matrix is Θ(n^4/3) (resp. Θ(n^3/2)). Relationships with point-line incidences in the plane, Bruhat order of permutations, and TP completability are also presented. We also examine the number and positionings of equal 2\textrm-by-2 minors in a 2\textrm-by-n TP matrix, and give a relationship between the location of equal 2\textrm-by-2 minors and outerplanar graphs.

Citations

• Forbidden paths and cycles in ordered graphs and matrices

Related

• On Totally Positive Matrices and Geometric Incidences
• On the number of CP factorizations of a completely positive matrix
• Permanental inequalities for totally positive matrices
• On Invariance Properties of Entries of Matrix Powers
• Arrangements of equal minors in the positive Grassmannian