  We develop a power series method for the nonequilibrium steady state of the
inhomogeneous one-dimensional totally asymmetric simple exclusion process
(TASEP) in contact with two particle reservoirs and with site-dependent hopping
rates in the bulk. The power series is performed in the entrance or exit rates
governing particle exchange with the reservoirs, and the corresponding particle
current is computed analytically up to the cubic term in the entry or exit
rate, respectively. We also show how to compute higher-order terms using
combinatorial objects known as Young tableaux. Our results address the long
outstanding problem of finding the exact nonequilibrium steady state of the
inhomogeneous TASEP. The findings are particularly relevant to the modelling of
mRNA translation in which the rate of translation initiation, corresponding to
the entrance rate in the TASEP, is typically small.
