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We consider the problem of building an expression for the partition sum incase of non-additivity (nano-sized objects), in the framework of Hill'snanothermodynamics. Having to use not the additivity concept leads to theproblem of building the generalised (Hill's) partition sum on base of thecombinatorial statistics averaging. Expressions for the energy of the objectare built on base of the vibron (algebraic) method, also using the author'smodel of size dependency of numbers of structural elements of the object. Next,the combinatorial-statistical averaging is introduced for the Hamiltonianmatrix elements. The final result is represented by the expressions for thenano-sized object's individual quantum states' energies and the partition sumallowing the factorisation.  Keywords: nanoparticles, nanothermodynamics, vibron model, algebraic method,Lie algebras, combinatorics, compositions

Partition sum in vibron model in case of non-additivity (nano-sized objects)