Geodesic
The birational Lalanne–Kreweras involution
Hopkins, Sam1 ; Joseph, Michael2
1 Department of Mathematics Howard University Washington DC USA
2 Department of Technology and Mathematics Dalton State College Dalton GA USA
Algebraic Combinatorics, Tome 5 (2022) no. 2, pp. 227-265.

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The Lalanne–Kreweras involution is an involution on the set of Dyck paths which combinatorially exhibits the symmetry of the number of valleys and major index statistics. We define piecewise-linear and birational extensions of the Lalanne–Kreweras involution. Actually, we show that the Lalanne–Kreweras involution is a special case of a more general operator, called rowvacuation, which acts on the antichains of any graded poset. Rowvacuation, like the closely related and more studied rowmotion operator, is a composition of toggles. We obtain the piecewise-linear and birational lifts of the Lalanne–Kreweras involution by using the piecewise-linear and birational toggles of Einstein and Propp. We show that the symmetry properties of the Lalanne–Kreweras involution extend to these piecewise-linear and birational lifts.

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DOI : 10.5802/alco.201
Classification : 05E18, 06A07, 05A19
Keywords: Lalanne–Kreweras involution, rowvacuation, rowmotion, toggles, piecewise-linear and birational lifts, homomesy

Hopkins, Sam 1 ; Joseph, Michael 2

1 Department of Mathematics Howard University Washington DC USA
2 Department of Technology and Mathematics Dalton State College Dalton GA USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Hopkins, Sam; Joseph, Michael. The birational Lalanne–Kreweras involution. Algebraic Combinatorics, Tome 5 (2022) no. 2, pp. 227-265. doi : 10.5802/alco.201. http://geodesic.mathdoc.fr/articles/10.5802/alco.201/

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