  A box-ball system (BBS) is a discrete dynamical system consisting of n balls
in an infinite strip of boxes. During each BBS move, the balls take turns
jumping to the first empty box, beginning with the smallest-numbered ball. The
one-line notation of a permutation can be used to define a BBS state. This
paper proves that the Robinson-Schensted (RS) recording tableau of a
permutation completely determines the dynamics of the box-ball system
containing the permutation.
  Every box-ball system eventually reaches steady state, decomposing into
solitons. We prove that the rightmost soliton is equal to the first row of the
RS insertion tableau and it is formed after at most one BBS move. This fact
helps us compute the number of BBS moves required to form the rest of the
solitons. First, we prove that if a permutation has an L-shaped soliton
decomposition then it reaches steady state after at most one BBS move.
Permutations with L-shaped soliton decompositions include noncrossing
involutions and column reading words. Second, we make partial progress on the
conjecture that every permutation on n objects reaches steady state after at
most n-3 BBS moves. Furthermore, we study the permutations whose soliton
decompositions are standard; we conjecture that they are closed under
consecutive pattern containment and that the RS recording tableaux belonging to
such permutations are counted by the Motzkin numbers.
