The birational Lalanne-Kreweras involution

TL;DR

This paper extends the Lalanne-Kreweras involution, originally defined on Dyck paths, to piecewise-linear and birational settings, revealing its connection to a broader operator called rowvacuation that acts on antichains of graded posets.

Contribution

It introduces birational and piecewise-linear versions of the Lalanne-Kreweras involution and links it to the general rowvacuation operator on graded posets.

Findings

Lalanne-Kreweras involution is a special case of rowvacuation.

Piecewise-linear and birational lifts preserve symmetry properties.

Connections established with toggle operations of Einstein and Propp.

Abstract

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.