A combinatorial interpretation of the scalar products of state vectors of integrable models

The representation of Bethe wave functions of certain integrable models via Schur functions allows one to apply the well-developed theory of symmetric functions to the calculation of thermal correlation functions. The algebraic relations arising in the calculation of scalar products and correlation functions are based on the Binet–Cauchy formula for the Schur functions. We provide a combinatorial interpretation of the formula for the scalar products of Bethe state vectors in terms of nests of self-avoiding lattice paths constituting so-called watermelon configurations. The proposed interpretation is, in turn, related to the enumeration of boxed plane partitions.