  The Hermite Normal Form (HNF) is a canonical representation of matrices over
any principal ideal domain. Over the integers, the distribution of the HNFs of
randomly looking matrices is far from uniform. The aim of this article is to
present an explicit computation of this distribution together with some
applications. More precisely, for integer matrices whose entries are upper
bounded in absolute value by a large bound, we compute the asymptotic number of
such matrices whose HNF has a prescribed diagonal structure. We apply these
results to the analysis of some procedures and algorithms whose dynamics depend
on the HNF of randomly looking integer matrices.
