We propose a computational approach for implementing discrete hidden semi-Markov chains. A discrete hidden semi-Markov chain is composed of a non-observable or hidden process which is a finite semi-Markov chain and a discrete observable process. Hidden semi-Markov chains possess both the flexibility of hidden Markov chains for approximating complex probability distributions and the flexibility of semi-Markov chains for representing temporal structures. Efficient algorithms for computing characteristic distributions organized according to the intensity, interval and counting points of view are described. The proposed computational approach in conjunction with statistical inference algorithms previously proposed makes discrete hidden semi-Markov chains a powerful model for the analysis of samples of non-stationary discrete sequences.