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For each fully commutative permutation, we construct a “boolean core,” which is the maximal boolean permutation in its principal order ideal under the right weak order. We partition the set of fully commutative permutations into the recently defined crowded and uncrowded elements, distinguished by whether or not their RSK insertion tableaux satisfy a sparsity condition. We show that a fully commutative element is uncrowded exactly when it shares the RSK insertion tableau with its boolean core. We present the dynamics of the right weak order on fully commutative permutations, with particular interest in when they change from uncrowded to crowded. In particular, we use consecutive permutation patterns and descents to characterize the minimal crowded elements under the right weak order. 

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. 

A box-ball system is a collection of discrete time states. At each state, we have a collection of countably many boxes with each integer from 1 to n assigned to a unique box; the remaining boxes are considered empty. A permutation on n objects gives a box-ball system state by assigning the permutation in one-line notation to the first n boxes. After a finite number of steps, the system will reach a so-called soliton decomposition which has an integer partition shape. We prove the following: if the soliton decomposition of a permutation is a standard Young tableau or if its shape coincides with its Robinson–Schensted (RS) partition, then its soliton decomposition and its RS insertion tableau are equal. We study the time required for a box-ball system to reach a steady state. We also generalize Fukuda’s single-carrier algorithm to algorithms with more than one carrier.