An Extension of an Inequality Involving Symmetric Products with an Application to Permanents

Let A be an nk × nk positive semi-definite symmetric matrix partitioned into blocks A^ij each of which is an n × n matrix. In 2] Mine states a conjecture of Marcus that per(A) ≥ per(G) where G is the k × k matrix per(A^ij)]. In this paper we prove a weaker inequality namely that per(A) ≥ (k!)^-1per(G).