A systematic study is performed of fully integrated particle multiplicities in central Au-Au and Pb-Pb collisions at beam momenta of 1.7A GeV/c, 11.6A GeV/c (Au-Au) and 158A GeV/c (Pb-Pb) by using a statistical-thermal model. The close similarity of the colliding systems makes it possible to study heavy ion collisions under definite initial conditions over a range of centre-of-mass energies covering more than one order of magnitude. In order to further study the behaviour of strangeness production, an updated study of Si-Au collisions at 14.6A GeV is also presented. The data analysis has been performed with two completely independent numerical algorithms giving closely consistent results. We conclude that a thermal model description of particle multiplicities, with additional strangeness suppression, is possible for each energy. The degree of chemical equilibrium of strange particles and the relative production of strange quarks with respect to u and d quarks are higher than in e + e − , pp and pp collisions at comparable and even at lower energies. The behaviour of strangeness production as a function of centre-of-mass energy and colliding system is presented and discussed. The average energy per hadron in the comoving frame is close to 1 GeV per hadron despite the fact that the energy increases more than 10-fold.