Tableau sequences, open diagrams, and Baxter families

Abstract

Walks on Young’s lattice of integer partitions encode many objects of algebraic and combinatorial interest. Chen et al. established connections between such walks and arc diagrams. We show that walks that start at ∅, end at a row shape, and only visit partitions of bounded height are in bijection with a new type of arc diagram — open diagrams. Remarkably, two subclasses of open diagrams are equinumerous with well known objects: standard Young tableaux of bounded height, and Baxter permutations. We give an explicit combinatorial bijection in the former case, and a generating function proof and new conjecture in the second case.