  We demonstrate here a series of exact mappings between particular cases of
four statistical physics models: equilibrium 1-dimensional lattice gas with
nearest-neighbor repulsion, $(1+1)$-dimensional combinatorial heap of pieces,
random walks on half-plane and totally asymmetric simple exclusion process
(TASEP) in one dimension (1D). In particular, we show that generating function
of a steady state of one-dimensional TASEP with open boundaries can be
interpreted as a quotient of partition functions of 1D hard-core lattice gases
with one adsorbing lattice site and negative fugacity. This result is based on
the combination of (i) a representation of the steady-state TASEP
configurations in terms of $(1+1)$-dimensional heaps of pieces and (ii) a
theorem connecting the partition function of $(1+1)$-dimensional heaps of
pieces with that of a single layer of pieces, which in this case is a 1D
hard-core lattice gas.
