  We introduce ultradiscrete tau functions associated with rigged
configurations for A^{(1)}_n. They satisfy an ultradiscrete version of the
Hirota bilinear equation and play a role analogous to a corner transfer matrix
for the box-ball system. As an application, we establish a piecewise linear
formula for the Kerov-Kirillov-Reshetikhin bijection in the combinatorial Bethe
ansatz. They also lead to general N-soliton solutions of the box-ball system.
