Promotion and rowmotion are bijections that arise in different combinatorial contexts. Promotion acts on linear extensions of posets — it was introduced by Schützenberger in the study of Young tableaux and has deep connections to representation theory. Rowmotion acts on order ideals of posets — it was studied by Brouwer and Schrijver and connects to toggle dynamics and piecewise-linear combinatorics. Both have been extended to many settings, but their relationship has been unclear outside specific cases.

Shigechi (arXiv:2603.17402) proves that on generalized Dyck paths in rational Catalan combinatorics, promotion and rowmotion are equivalent. The same bijection, viewed from two different combinatorial frameworks, produces the same dynamical system.

The equivalence passes through classical operations. Promotion on Dyck paths corresponds to rotation — a geometric operation that shifts the path cyclically. Rowmotion corresponds to the Kreweras complement — an algebraic operation on noncrossing partitions. That rotation and the Kreweras complement produce the same dynamics on rational Dyck paths was known in special cases; Shigechi proves it in full generality using a novel RSK-type correspondence through Dyck tilings.

Dyck tilings — arrangements of Dyck paths that tile a region in the plane — serve as the bridge. The RSK correspondence for ordinary partitions connects tableaux operations to matrix operations; this Dyck tiling correspondence connects promotion-type operations to rowmotion-type operations. The tiling is the common language that reveals the operations as the same.

Two traditions, two notations, two communities — one bijection. The equivalence is not an analogy but an identity: promotion on these paths IS rowmotion, and the proof shows exactly how the identification works at every step.