The birational Lalanne–Kreweras involution

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