Runs and RSK tableaux of boolean permutations

TL;DR

This paper introduces a canonical reduced word for boolean permutations, links it to RSK tableaux, characterizes and counts tableaux associated with boolean permutations, and explores how the 'run' statistic affects RSK shapes.

Contribution

It provides a new method to construct and interpret RSK tableaux for boolean permutations and generalizes the influence of the 'run' statistic to all permutations.

Findings

RSK tableaux can be directly read from the canonical reduced word.

Characterization and enumeration of tableaux corresponding to boolean permutations.

The 'run' statistic affects RSK tableau shape for all permutations.

Abstract

We define and construct the "canonical reduced word" of a boolean permutation, and show that the RSK tableaux for that permutation can be read off directly from this reduced word. We also describe those tableaux that can correspond to boolean permutations, and enumerate them. In addition, we generalize a result of Mazorchuk and Tenner, showing that the "run" statistic influences the shape of the RSK tableau of arbitrary permutations, not just of those that are boolean.