Séminaire Lotharingien de Combinatoire, 95B.35 (2026), 12 pp.
Colin Defant, Yuhan Jiang, Rene Marczinzik, Adrien Segovia, David E Speyer, Hugh Thomas and Nathan Williams*
Rowmotion and Echelonmotion
Abstract.
Given a linear extension &sigmaL; of a finite poset R, we
consider the permutation matrix indexing the Bruhat cell containing
the Cartan matrix of R with respect to σ. This yields a
bijection Echσ : R → R that
we call echelonmotion; it is the inverse of the Coxeter permutation studied by Klá,sz, Marczinzik, and Thomas. Those authors proved that echelonmotion agrees with rowmotion when R is a distributive lattice. We generalize this result to semidistributive lattices. In addition, we prove that every trim lattice has a linear extension with respect to which echelonmotion agrees with rowmotion. We also show that echelonmotion on an Eulerian poset (with respect to any linear extension) is an involution. Finally, we initiate the study of echelon-independent posets, which are posets for which echelonmotion is independent of the chosen linear extension. We prove that a lattice is echelon-independent if and only if it is semidistributive. Moreover, we show that echelon-independent connected posets are bounded and have semidistributive MacNeille completions.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
The following versions are available: