Symmetry of Narayana Numbers and Rowvacuation of Root Posets
Forum of Mathematics, Sigma, Tome 9 (2021)
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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.
@article{10_1017_fms_2021_47,
author = {Colin Defant and Sam Hopkins},
title = {Symmetry of {Narayana} {Numbers} and {Rowvacuation} of {Root} {Posets}},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {9},
year = {2021},
doi = {10.1017/fms.2021.47},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2021.47/}
}
Colin Defant; Sam Hopkins. Symmetry of Narayana Numbers and Rowvacuation of Root Posets. Forum of Mathematics, Sigma, Tome 9 (2021). doi: 10.1017/fms.2021.47
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