[en] We identify the algorithm for constructing steady states of the n-species totally asymmetric simple exclusion process (TASEP) on an L site periodic chain by Ferrari and Martin with a composition of combinatorial R for the quantum affine algebra $U_q({\stackrel{^<!--\; ^\; -->}{sl}}_L)$ in crystal base theory. Based on this connection and the factorized form of the R matrix derived recently from the tetrahedron equation, we establish a new matrix product formula for the steady state of the TASEP, which is expressed in terms of corner transfer matrices of the q-oscillator valued five-vertex model at q = 0. (fast track communication)