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 -