  We define and study multi-colored dimer models on a segment and on a circle.
The multivariate generating functions for the dimer models satisfy the
recurrence relations similar to the one for Fibonacci numbers. We give closed
formulae for the generating functions. We show that, in the large size limit
with specializations of the formal variables, the generating functions exhibit
the summations appearing in generalized Rogers--Ramanujan identities. Further,
the generating functions of the dimer models have infinite product formulae for
general values of formal variables in the large size limit. These formulae are
generalizations of Rogers--Ramanujan identities for multi variables. We also
give other several specializations which exhibit simple combinatorial formulae.
The analysis of the correlation functions, which we call emptiness formation
probabilities and moments, leads to the application of the formal power series
associated to the Dyck, Motzkin and Schr\"oder paths to the generating
functions for the dimer models. We give descriptions of the generating
functions of finite size in terms of these combinatorial objects, Dyck and
Motzkin paths with statistics. We have three additional results. First, the
convoluted generating functions for Fibonacci, Catalan and Motzkin numbers are
shown to be expressed as generating functions of Fibonacci, Dyck and Motzkin
words with the weights given by binomial coefficients. The second one is a
weight preserving correspondence between a Motzkin path and a set of Dyck
paths. The third one is a connection of the generating functions for the dimer
models to the generating functions of independent sets of special classes of
graphs.
