A box-ball system is a discrete dynamical system whose dynamics come from the balls jumping according to certain rules. A permutation on \(n\) objects gives a box-ball system state by assigning its one-line notation to \(n\) consecutive boxes. After a finite number of steps, a box-ball system will reach a steady state. From any steady state, we can construct a tableau called the soliton decomposition of the box-ball system. We prove that if the soliton decomposition of a permutation \(w\) is a standard tableau or if its shape coincides with the Robinson-Schensted (RS) partition of \(w\), then the soliton decomposition of \(w\) and the RS insertion tableau of \(w\) are equal. We also use row reading words, Knuth moves, RS recording tableaux, and a localized version of Greene's theorem (proven recently by Lewis, Lyu, Pylyavskyy, and Sen) to study various properties of a box-ball system.

Mathematics Subject Classifications: 05A05, 05A17, 37B15

Keywords: Permutations, box-ball systems, soliton cellular automata, Young tableaux, Robinson-Schensted-Knuth correspondence, Greene's theorem, Knuth equivalence