Generating functions of Box and Ball System

Generating functions of Box and Ball System (BBS) are defined and studied. When the number of balls is finite, we show that the generating function is a rational function. When there are infinitely many balls, we conjecture that the generating function is rational if and only if the BBS is semi-periodic. We prove the conjecture in a special case. We also study the generating function of the BBS with a limited cart, including semi-periodic cases.