Discoveries / combinatorics

Conjectures of Hopkins, Sagan-Wilson and Defant et al. on Lattices, Parking Functions and the Plactic Monoid

A collection of open problems from the algebraic and enumerative combinatorics literature, resolved in one paper: a conjecture of Defant, Jiang, Marczinzik, Segovia, Speyer, Thomas and Williams on the echelonmotion operator on modular lattices, which also yields a new algebraic bijective proof of Dilworth's theorem; conjectures of Hopkins on parking function statistics studied by Stanley and Yin; and two conjectures of Sagan and Wilson on centralizers in the plactic monoid.