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Title: Symmetry of Narayana Numbers and Rowvacuation 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 – that is, 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.  more » « less
Award ID(s):
1802920
PAR ID:
10353968
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Forum of Mathematics, Sigma
Volume:
9
ISSN:
2050-5094
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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