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