In classical statistical physics, the challenging combinatorial enumeration of the configurations of a system subject to hard constraints (microcanonical ensemble) is mapped to a mathematically easier calculation where the constraints are softened (canonical ensemble). However, the mapping is exact only as the system size diverges and if the property of ensemble equivalence (EE) holds. For finite systems, or when EE breaks down, statistical physics is believed to provide no answer to the combinatorial problem.
In contrast with this expectation, here we establish exact relationships connecting the two conjugate ensembles in full generality and for an arbitrary number of constraints, even for finite system size and when EE does not hold. We also show that the ensembles are asymptotically related through the matrix of canonical (co)variances of the constraints. These relationships restore the possibility of enumerating microcanonical configurations via canonical probabilities, thus reconnecting statistical physics and combinatorics in realms where they were believed to be no longer in mutual correspondence. Moreover, they highlight a completely new mechanism, recently observed in specific models but so far unexplained on general grounds, whereby EE can be broken even in absence of phase transitions: namely, the presence of a diverging number of constraints. As a concrete application, we show that ensembles of graphs with local constraints, widely used in network science but hard to enumerate, break EE through the new mechanism; yet, our approach yields an asymptotic enumeration of their configurations, using only canonical quantities.