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For a Weyl group $W$ of rank $r$, the $W$-Catalan number is the number ofantichains of the poset of positive roots, and the $W$-Narayana numbers refinethe $W$-Catalan number by keeping track of the cardinalities of theseantichains. The $W$-Narayana numbers are symmetric, i.e., the number ofantichains of cardinality $k$ is the same as the number of cardinality $r-k$.However, this symmetry is far from obvious. Panyushev posed the problem ofdefining an involution on root poset antichains that exhibits the symmetry ofthe $W$-Narayana numbers.  Rowmotion and rowvacuation are two related operators, defined as compositionsof "toggles," that give a dihedral action on the set of antichains of anyranked poset. Rowmotion acting on root posets has been the subject of asignificant amount of research in the recent past. We prove that for the rootposets of classical types, rowvacuation is Panyushev's desired involution.

Symmetry of Narayana numbers and rowvacuation of root posets