Box-ball systems and RSK recording tableaux
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 coincide with standard tableaux; 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.