  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 several
methods. Recently, a new linearization method, called the seat number
configuration, is introduced by arXiv:2301.00132. The aim of this paper is
fourfold. 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 map
induces a shift operator of the seat number configuration. Second, we show that
the $k$-skip map is a natural generalization of the $10$-elimination, which was
originally introduced by [MIT] to solve the initial value problem of the
periodic BBS. Third, we generalize the notions and results of the seat number
configuration and the $k$-skip map for the BBS on the whole-line. Finally, we
investigate the distribution of $\Psi_{k}(\eta), \eta \in \Omega$ when the
distribution of $\eta$ belongs to a certain class of invariant measures of the
BBS introduced by [FG]. As an application of the above results, we obtain the
long-time behavior of the integrated current of $\Psi_{k}(\eta)$ with Markov
stationary initial distributions.
