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

The box-ball system (BBS) is a soliton cellular automaton introduced in [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 in [MSSS]. The aim of this paper is fourfold. First, we introduce the $k$-skip map ${\mathrm{\Psi }}_k:\mathrm{\Omega }\to \mathrm{\Omega }$, where $\mathrm{\Omega }$ 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 in [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 ${\mathrm{\Psi }}_k(\eta ),\eta \in \mathrm{\Omega }$ when the distribution of $\eta$ belongs to a certain class of invariant measures of the BBS introduced in [FG]. As an application of the above results, we compute the expectation of the carrier with seat numbers.