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:
Classification: 05E18, 06A07, 05A19
Keywords: Lalanne–Kreweras involution, rowvacuation, rowmotion, toggles, piecewise-linear and birational lifts, homomesy
Author's affiliations:
@article{ALCO_2022__5_2_227_0, author = {Hopkins, Sam and Joseph, Michael}, title = {The birational {Lalanne{\textendash}Kreweras} involution}, journal = {Algebraic Combinatorics}, pages = {227--265}, publisher = {The Combinatorics Consortium}, volume = {5}, number = {2}, year = {2022}, doi = {10.5802/alco.201}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.201/} }
TY - JOUR TI - The birational Lalanne–Kreweras involution JO - Algebraic Combinatorics PY - 2022 DA - 2022/// SP - 227 EP - 265 VL - 5 IS - 2 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.201/ UR - https://doi.org/10.5802/alco.201 DO - 10.5802/alco.201 LA - en ID - ALCO_2022__5_2_227_0 ER -
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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