Symmetry of Narayana Numbers and Rowvacuation of Root Posets

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