Tower Tableaux

TL;DR

This paper introduces tower diagrams, a new combinatorial object, along with an algorithm to slide words onto these diagrams, establishing a bijection with permutations and a test for reducibility.

Contribution

The paper presents the novel concept of tower diagrams, an algorithm for sliding words, and a bijection with permutations, advancing combinatorial and permutation theory.

Findings

Algorithm is well-defined only for reduced words

Bijection between tower diagrams and permutations established

Algorithm serves as a reducibility test

Abstract

We introduce a new combinatorial object called tower diagrams and prove fundamental properties of these objects. We also introduce an algorithm that allows us to slide words to tower diagrams. We show that the algorithm is well-defined only for reduced words which makes the algorithm a test for reducibility. Using the algorithm, a bijection between tower diagrams and finite permutations is obtained and it is shown that this bijection specializes to a bijection between certain labellings of a given tower diagram and reduced expressions of the corresponding permutation.