  In 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 when the size of the system is infinite and if the property of ensemble
equivalence (EE), i.e. the asymptotic identity of canonical and microcanonical
large deviations, holds. For finite systems, or when EE breaks down,
statistical physics is currently believed to provide no answer to the
combinatorial problem. In contrast with this expectation, here we establish
exact relationships connecting conjugate ensembles in full generality, even for
finite system size and when EE does not hold. We also show that in the
thermodynamic limit the ensembles are directly related through the matrix of
canonical (co)variances of the constraints, plus a correction term that
survives only if this matrix has an infinite number of finite eigenvalues.
These new 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.
