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A birational lifting of the Lalanneâ Kreweras involution on Dyck paths Joseph, Michael


The Lalanneâ Kreweras involution (LK) on Dyck paths yields a bijective proof of the symmetry of two statistics: the number of valleys and the major index. Equivalently, this involution can be considered on the set of antichains of the type A root poset, on which rowmotion and LK together generate a dihedral action (as first discovered by Panyushev). Piecewise-linear and birational rowmotion were first defined by Einstein and Propp. Moving further in this direction, we define an analogue of the LK involution to the piecewise-linear and birational realms. In fact, LK is a special case of a more general action, rowvacuation, an involution that can be defined on any finite graded poset where it forms a dihedral action with rowmotion. We will explain that the symmetry properties of the number of valleys and the major index also lift to the higher realms. In this process, we have discovered more refined homomesies for LK, and we will explain how certain statistics which are homomesic under rowvacuation are also homomesic under rowmotion. This is joint work with Sam Hopkins.

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