On some permanental conjectures

Abstract

Let Ω n denote the set of all n×n doubly-stochastic matrices and let σ k (A) be the sum of all subpermanents of order k of matrix A. We prove the Holens-Dokovic conjecture for k=4 and n≥5. Namely, Let be th set of all matrices from Ω n with zero main diagonal and let be the matrix from with 1/(n−1) in its off-diagonal positions. We prove the following for 2≤k≤4 and n≥k+1: for any A consequence of the last inequality and also of a result of D. London and H. Minc [8] is the inequality .