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On two-toned tilings and $(m,n)$-words

Henri Mühle

TL;DR

The paper tackles the problem of relating two combinatorial models that count the same coefficients in $x^n$ for $((1-x)/(1-2x))^{m+1}$ by constructing an explicit bijection between $(m,n)$-words and two-toned tilings of length $m+n$. It provides a concrete forward map that decomposes $(m,n)$-words into segments separated by runs of $(m+1)$'s and associates these to blue strips and red blocks in a tiling, along with an explicit inverse map. The bijection confirms a tight correspondence between the two models and enables transferring structural insights between them. This unifies perspectives on Hochschild polytope-related combinatorics and opens avenues to study induced order structures, such as core label order, on the tiling side.

Abstract

In this article, we describe an explicit bijection between the set of $(m,n)$-words as defined by Pilaud and Poliakova and the set of of two-toned tilings of a strip of length $m+n$.

On two-toned tilings and $(m,n)$-words

TL;DR

The paper tackles the problem of relating two combinatorial models that count the same coefficients in for by constructing an explicit bijection between -words and two-toned tilings of length . It provides a concrete forward map that decomposes -words into segments separated by runs of 's and associates these to blue strips and red blocks in a tiling, along with an explicit inverse map. The bijection confirms a tight correspondence between the two models and enables transferring structural insights between them. This unifies perspectives on Hochschild polytope-related combinatorics and opens avenues to study induced order structures, such as core label order, on the tiling side.

Abstract

In this article, we describe an explicit bijection between the set of -words as defined by Pilaud and Poliakova and the set of of two-toned tilings of a strip of length .

Paper Structure

This paper contains 8 sections, 7 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Basics
  3. $(m,n)$-Words
  4. Two-toned tilings
  5. A bijection between $(m,n)$-words and two-toned tilings
  6. From $(m,n)$-words to two-toned tilings
  7. From two-toned tilings to $(m,n)$-words
  8. Possible next steps

Key Result

Lemma 2.1

For $m\geq 0$, $n\geq 1$, the number of $(m,n)$-words is

Theorems & Definitions (15)

  • Lemma 2.1: muehle24combinatorics
  • Example 2.2
  • Lemma 2.3: benjamin11combinatorics
  • Lemma 2.4: davis20further
  • Example 2.5
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Example 3.4
  • ...and 5 more