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Box-ball systems and RSK recording tableaux

Marisa Cofie, Olivia Fugikawa, Emily Gunawan, Madelyn Stewart, David Zeng

TL;DR

The paper establishes a deep link between box-ball systems (BBS) and the Robinson--Schensted (RS) correspondence by proving that the RS recording tableau $Q(w)$ completely determines the BBS dynamics of a permutation, including steady-state time and the soliton partition. It shows that the rightmost soliton equals the first row of the RS insertion tableau and forms after at most one BBS move, enabling analysis of subsequent solitons; it also introduces the notion of good recording tableaux and analyzes L-shaped soliton decompositions, noncrossing/nested involutions, and column reading words. The work leverages dual Knuth equivalence and the carrier algorithm to relate BBS moves with RS insertions and to prove that $Q(w)$ fixes the entire time evolution of a BBS. It also provides partial results toward the maximum steady-state time conjecture, constructs tableaux with a range of steady-state times, and links these combinatorics to Motzkin numbers and pattern containment questions. Overall, the paper advances a precise, algorithmic bridge between integrable discrete dynamics (BBS) and classical tableau theory, with open directions in pattern avoidance and enumeration.

Abstract

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.

Box-ball systems and RSK recording tableaux

TL;DR

The paper establishes a deep link between box-ball systems (BBS) and the Robinson--Schensted (RS) correspondence by proving that the RS recording tableau completely determines the BBS dynamics of a permutation, including steady-state time and the soliton partition. It shows that the rightmost soliton equals the first row of the RS insertion tableau and forms after at most one BBS move, enabling analysis of subsequent solitons; it also introduces the notion of good recording tableaux and analyzes L-shaped soliton decompositions, noncrossing/nested involutions, and column reading words. The work leverages dual Knuth equivalence and the carrier algorithm to relate BBS moves with RS insertions and to prove that fixes the entire time evolution of a BBS. It also provides partial results toward the maximum steady-state time conjecture, constructs tableaux with a range of steady-state times, and links these combinatorics to Motzkin numbers and pattern containment questions. Overall, the paper advances a precise, algorithmic bridge between integrable discrete dynamics (BBS) and classical tableau theory, with open directions in pattern avoidance and enumeration.

Abstract

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.
Paper Structure (32 sections, 35 theorems, 58 equations, 2 figures)

This paper contains 32 sections, 35 theorems, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Box-ball systems
  3. Robinson--Schensted tableaux
  4. RS recording tableaux
  5. First solitons and steady-state times
  6. Steady State
  7. Increasing decomposition and steady state
  8. Configuration array and steady state
  9. A localized version of Schensted's theorem for box-ball systems
  10. Schensted's theorem and RS partition
  11. Localized Schensted's theorem and BBS soliton partition
  12. Recording tableau determines BBS dynamics
  13. Dual Knuth equivalence
  14. BBS moves preserve dual Knuth equivalence
  15. Q determines the steady-state time of a permutation
  16. ...and 17 more sections

Key Result

Theorem A

If $\pi, w$ are permutations such that $\mathop{\mathrm{Q}}\nolimits(\pi) = \mathop{\mathrm{Q}}\nolimits(w)$, then the following holds.

Figures (2)

  • Figure 1: One BBS move from $\dots ee452361eeee\dots$ to $\dots eeee45e2136e \dots$
  • Figure 2: A box-ball system with the permutation $452361$ at $t=0$

Theorems & Definitions (87)

  • Theorem A
  • Theorem B: Theorem \ref{['thm:first soliton created after 1 BBM']}
  • Theorem C: Theorem \ref{['thm: L-shaped SD SST']}
  • Theorem D: Theorem \ref{['thm:max SST of T with sh(n-3,2,1)']}
  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5: Steady-state characterization using ID
  • Proposition 2.6: Steady-state characterization using CA
  • ...and 77 more