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The box-ball system (BBS) is a soliton cellular automaton introduced by [TS],and it is known that the dynamics of the BBS can be linearized by severalmethods. Recently, a new linearization method, called the seat numberconfiguration, is introduced by arXiv:2301.00132. The aim of this paper isfourfold. First, we introduce the $k$-skip map $\Psi_{k} : \Omega \to \Omega$,where $\Omega$ is the state space of the BBS, and show that the $k$-skip mapinduces a shift operator of the seat number configuration. Second, we show thatthe $k$-skip map is a natural generalization of the $10$-elimination, which wasoriginally introduced by [MIT] to solve the initial value problem of theperiodic BBS. Third, we generalize the notions and results of the seat numberconfiguration and the $k$-skip map for the BBS on the whole-line. Finally, weinvestigate the distribution of $\Psi_{k}(\eta), \eta \in \Omega$ when thedistribution of $\eta$ belongs to a certain class of invariant measures of theBBS introduced by [FG]. As an application of the above results, we obtain thelong-time behavior of the integrated current of $\Psi_{k}(\eta)$ with Markovstationary initial distributions.

Seat number configuration of the box-ball system, and its relation to the 10-elimination and invariant measures