The aim of this manuscript is to understand the dynamics products nonnegative matrices. We extend a well known consequence Perron-Frobenius theorem on periodic points matrix finitely many matrices associated word and later word, possibly infinite length.

In this paper we show that finite rings for which the extension theorem of MacWilliams is valid for Hamming weight must necessarily be Frobenius. This result makes use of a strategy of Dinh and López-Permouth.

For any set representation (permutation representation) of the symmetric group Sn, we give combinatorial interpretation for coefficients of its Frobenius character expanded in the basis of monomial symmetric functions.