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Walks on tiled boards

László Németh

Abstract

Several articles deal with tilings with various shapes, and also a very frequent type of combinatorics is to examine the walks on graphs or on grids. We combine these two things and give the numbers of the shortest walks crossing the tiled $(1\times n)$ and $(2\times n)$ square grids by covering them with squares and dominoes. We describe these numbers not only recursively, but also as rational polynomial linear combinations of Fibonacci numbers.

Walks on tiled boards

Abstract

Several articles deal with tilings with various shapes, and also a very frequent type of combinatorics is to examine the walks on graphs or on grids. We combine these two things and give the numbers of the shortest walks crossing the tiled and square grids by covering them with squares and dominoes. We describe these numbers not only recursively, but also as rational polynomial linear combinations of Fibonacci numbers.
Paper Structure (4 sections, 7 theorems, 62 equations, 11 figures, 2 tables)

This paper contains 4 sections, 7 theorems, 62 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Tiling and walking on --board
  3. Tiling and walking on --board
  4. Tiling and walking with only dominoes

Key Result

Theorem 1

The tiling-walking sequence $(v_n)_{n=0}^\infty$ of the $(1\times n)$-board with squares and dominoes has the recurrence relation where the initial values are $v_0=1$, $v_1=2$ (see A001629 in OEISoeis).

Figures (11)

  • Figure 1: Examples for walks on tiled boards
  • Figure 2: Walks on $2\times3$-board with a given tiling
  • Figure 3: All the walks on the tiled boards $1\times 0$, $1\times 1$, and $1\times 2$
  • Figure 4: Tiling and walks on $(1\times n)$-board with recurrence
  • Figure 5: Tiling with exactly $k$ dominoes on $(1\times n)$-board
  • ...and 6 more figures

Theorems & Definitions (10)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Lemma 2.1: Benjamin and Quinn BQ
  • proof : Proof of Corollary \ref{['cor:main_1xn']}
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof