On the distribution of permanents of matrices over finite fields

For a prime power q, we study the distribution of permanents of n×n matrices over a finite field Fq of q elements. We show that if A is a sufficient large subset of Fq then the set of permanents of n×n matrices with entries in A covers all (or almost) . When q=p is a prime, and A is a subinterval of [0,p−1] of cardinality |A|≫p1/2logp, we show that the number of matrices with entries in A having permanent t is asymptotically close to the expected value. We also study this problem in more general settings.