  We introduce a new type of lattice path, called brick-wall lattice path, and
we derive a formula which counts the number of paths on these lattices imposing
certain restrictions on the Cartesian plane. Connections to the Fibonacci
sequence, as well as to other sequences of numbers, are given. Finally, we use
these brick-wall lattice paths to determine the first two non-zero coefficients
of the reliability polynomials associated with particular two-terminal networks
known as hammocks.
