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We define and study multi-colored dimer models on a segment and on a circle.The multivariate generating functions for the dimer models satisfy therecurrence relations similar to the one for Fibonacci numbers. We give closedformulae for the generating functions. We show that, in the large size limitwith specializations of the formal variables, the generating functions exhibitthe summations appearing in generalized Rogers--Ramanujan identities. Further,the generating functions of the dimer models have infinite product formulae forgeneral values of formal variables in the large size limit. These formulae aregeneralizations of Rogers--Ramanujan identities for multi variables. We alsogive other several specializations which exhibit simple combinatorial formulae.The analysis of the correlation functions, which we call emptiness formationprobabilities and moments, leads to the application of the formal power seriesassociated to the Dyck, Motzkin and Schr\"oder paths to the generatingfunctions for the dimer models. We give descriptions of the generatingfunctions of finite size in terms of these combinatorial objects, Dyck andMotzkin paths with statistics. We have three additional results. First, theconvoluted generating functions for Fibonacci, Catalan and Motzkin numbers areshown to be expressed as generating functions of Fibonacci, Dyck and Motzkinwords with the weights given by binomial coefficients. The second one is aweight preserving correspondence between a Motzkin path and a set of Dyckpaths. The third one is a connection of the generating functions for the dimermodels to the generating functions of independent sets of special classes ofgraphs.

Multi-colored dimer models in one-dimension: lattice paths and generalized Rogers--Ramanujan identities