Symmetry of Narayana numbers and rowvacuation of root posets

TL;DR

This paper proves that for classical root posets, rowvacuation acts as the involution that explains the symmetry of Narayana numbers, linking combinatorial symmetry with algebraic operators.

Contribution

It establishes that rowvacuation provides the involution needed to explain Narayana number symmetry in classical root posets.

Findings

Rowvacuation acts as the involution for classical root posets.

Symmetry of Narayana numbers is explained via rowvacuation.

Provides new insight into the combinatorial structure of root posets.

Abstract

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.