Department of Mathematics

Combinatorics and Graph Theory

  •  Emily Gunawan, Oklahoma University
  •  Box-ball systems and Robinson-Schensted-Knuth tableaux
  •  09/08/2021
  •  3:00 PM - 3:50 PM
  •  Online (virtual meeting) (Virtual Meeting Link)
  •  Bruce E Sagan (bsagan@msu.edu)

The Robinson-Schensted (RS) correspondence is a famous bijection between permutations and pairs (P,Q) of standard tableaux of the same shape, called the RS partition. The RS partition and its conjugate record certain permutation statistics called Greene’s theorem statistics. A box-ball system is a discrete dynamical system which can be thought of as a collection of time states. A permutation on n objects gives a box-ball system state by assigning its one-line notation to n consecutive boxes. After a finite number of steps, a box-ball system will reach a steady state. From any steady state, we can construct a tableau (not necessarily standard) called the soliton decomposition. The shape of the soliton decomposition is called the BBS partition. An exciting discovery (made in 2019 by Lewis, Lyu, Pylyavskyy, and Sen) is that the BBS partition and its conjugate record a localized version of Greene’s theorem statistics. We will discuss a few new results: (1) The Q tableau of a permutation completely determines the dynamics of the corresponding box-ball system. (2) The permutations whose BBS partitions are L-shaped have steady-state time at most 1. This large class of permutations include column reading words and noncrossing involutions. (3) If the soliton decomposition of a permutation is a standard tableau or if its BBS partition coincides with its RS partition, then its soliton decomposition and its P tableau are equal. (4) Finally, we study the permutations whose P tableaux and soliton decompositions coincide and refer to them as “good". These “good” permutations are closed under consecutive pattern containment. Furthermore, we conjecture that the “good” Q tableaux are counted by the Motzkin numbers. This talk is based on REU projects with Ben Drucker, Eli Garcia, Aubrey Rumbolt, Rose Silver (UConn Math REU 2020) and Marisa Cofie, Olivia Fugikawa, Madelyn Stewart, David Zeng (SUMRY 2021).

 

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