Séminaire Lotharingien de Combinatoire, 85B.14 (2021), 12 pp.

Ben Drucker, Eli Garcia, Emily Gunawan and Rose Silver

Box-ball Systems and RSK Tableaux

Abstract. 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.

Received: December 1, 2020. Accepted: March 1, 2021. Final version: April 29, 2021.

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