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

Combinatorics