TY - JOUR
T1 - On a conjecture concerning the Bruhat order
AU - Fernandes, Rosário
AU - da Cruz, Henrique
AU - Salomão, Domingos
N1 - partially supported by the Fundacao para a Ciencia e a Tecnologia through the project UIBD/MAT/00297/2020.
project UID/MAT/00212/2019.
PY - 2020
Y1 - 2020
N2 - Let R and S be two sequences of positive integers in nonincreasing order having the same sum. Let A(R,S) be the class of all (0,1)-matrices with row sum vector R and column sum vector S. If A(R,S) is nonempty, an inversion in AεA(R,S) consists of two entries of A equal to 1, one of them is located to the top-right of the other. Let γ(A) be the total number of inversions in A. The Bruhat order is a partial order defined on A(R,S) and denoted by ≤ . In this paper, we prove the conjecture:“If A,CεA(R,S), A≠C and A≤C then γ(A)<γ(C) ”.
AB - Let R and S be two sequences of positive integers in nonincreasing order having the same sum. Let A(R,S) be the class of all (0,1)-matrices with row sum vector R and column sum vector S. If A(R,S) is nonempty, an inversion in AεA(R,S) consists of two entries of A equal to 1, one of them is located to the top-right of the other. Let γ(A) be the total number of inversions in A. The Bruhat order is a partial order defined on A(R,S) and denoted by ≤ . In this paper, we prove the conjecture:“If A,CεA(R,S), A≠C and A≤C then γ(A)<γ(C) ”.
U2 - 10.1016/j.laa.2020.04.015
DO - 10.1016/j.laa.2020.04.015
M3 - Article
SN - 0024-3795
VL - 600
SP - 82
EP - 95
JO - Linear Algebra and its Applications
JF - Linear Algebra and its Applications
ER -