Tableau sequences, open diagrams, and Baxter families

TL;DR

This paper introduces open diagrams as a new combinatorial object linked to walks on Young's lattice, establishing bijections with standard Young tableaux and Baxter permutations, thus connecting algebraic structures with well-known combinatorial classes.

Contribution

It defines open diagrams and proves their bijection with certain Young's lattice walks, revealing new links between algebraic and combinatorial objects.

Findings

Open diagrams are in bijection with certain Young's lattice walks.

Two subclasses of open diagrams correspond to standard Young tableaux and Baxter permutations.

Explicit bijections are provided for the case of standard Young tableaux.

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 $\varnothing$, 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.