# ALGEBRAIC COMBINATORICS

The birational Lalanne–Kreweras involution
Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 227-265.

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.

Revised:
Accepted:
Published online:
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
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Hopkins, Sam; Joseph, Michael. The birational Lalanne–Kreweras involution. Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 227-265. doi : 10.5802/alco.201. https://alco.centre-mersenne.org/articles/10.5802/alco.201/

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