论文标题
盒球系统和RSK记录tableaux
Box-ball systems and RSK recording tableaux
论文作者
论文摘要
盒球系统（BBS）是一个离散的动力系统，该系统由无限条带中的N球组成。在每个BB的移动过程中，球转弯跳到第一个空盒子，从最小的球开始。置换的单行表示法可用于定义BBS状态。本文证明，置换置换的Robinson-Schensted（RS）记录图表完全决定了包含排列的盒球系统的动力学。 每个盒球系统最终都达到稳态，将其分解为孤子。我们证明，最右边的唯一唯一等于RS插入图表的第一行，并且最多在一个BBS移动后形成。这一事实有助于我们计算形成唯一的其余部分所需的BBS动作数量。首先，我们证明，如果置换具有L形的孤子分解，那么它最多在一个BBS移动后达到稳态。用L形孤子分解的排列包括非交叉的互动和列读单词。其次，我们对猜想做出了部分进步，即N对象上的每个置换量最多在N-3 BBS移动后都达到稳态。此外，我们研究了孤子分解是标准配置的排列；我们猜想它们在连续模式遏制下被关闭，并且属于此类排列的RS记录tableaux由Motzkin数字计数。
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.
