The nonnegative inverse elementary divisors problem (NIEDP) is the problem of finding conditions for the existence of an n x n entrywise nonnegative matrix A with prescribed elementary divisors. We consider the case in which the solution matrix A is required to be persymmetric. Persymmetric matrices are common in physical sciences and engineering. They arise, for instance, in the control of mechanical and electric vibrations. In this paper, we solve the NIEDP for n x n matrices assuming that (i) there exists a partition of the given list Lambda = {lambda(1),...,lambda(n)} in sublists Lambda(k), along with suitably chosen Perron eigenvalues, which are realizable by nonnegative matrices A(k) with certain of the prescribed elementary divisors, and (ii) a nonnegative persymmetric matrix exists with diagonal entries being the Perron eigenvalues of the matrices A(k), with certain of the prescribed elementary divisors. Our results generate an algorithmic procedure to compute the structured solution matrix. (C) 2015 Elsevier Inc. All rights reserved.