The Inequalities of Merris and Foregger for Permanents.

Conjectures on permanents are well-known unsettled conjectures in linear algebra. Let A be an n × n matrix and S n be the symmetric group on n element set. The permanent of A is defined as per A = ∑ σ ∈ S n ∏ i = 1 n a i σ (i). The Merris conjectured that for all n × n doubly stochastic matrices (denoted by Ω n ), n per A ≥ min 1 ≤ i ≤ n ∑ j = 1 n per A (j | i) , where A (j | i) denotes the matrix obtained from A by deleting the jth row and ith column. Foregger raised a question whether per (t J n + (1 − t) A) ≤ per A for 0 ≤ t ≤ n n − 1 and for all A ∈ Ω n , where J n is a doubly stochastic matrix with each entry 1 n . The Merris conjecture is one of the well-known conjectures on permanents. This conjecture is still open for n ≥ 4 . In this paper, we prove the Merris inequality for some classes of matrices. We use the sub permanent inequalities to prove our results. Foregger's inequality is also one of the well-known inequalities on permanents, and it is not yet proved for n ≥ 5 . Using the concepts of elementary symmetric function and subpermanents, we prove the Foregger's inequality for n = 5 in [0.25, 0.6248]. Let σ k (A) be the sum of all subpermanents of order k. Holens and Dokovic proposed a conjecture (Holen–Dokovic conjecture), which states that if A ∈ Ω n , A ≠ J n and k is an integer, 1 ≤ k ≤ n , then σ k (A) ≥ (n − k + 1) 2 n k σ k − 1 (A) . In this paper, we disprove the conjecture for n = k = 4 . [ABSTRACT FROM AUTHOR]