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Words with Repeated Letters in a Grid

Zachary Halberstam, Carl Schildkraut

TL;DR

The paper extends the study of word occurrences in large d-dimensional grids beyond distinct-letter words to those with repeated letters, introducing C_d(w) and the concept of w-extremal grids. It develops a multi-faceted methodology combining additive formulations, combinatorial reductions, local linear programming bounds, and Fourier analysis to establish sharp results in one and two dimensions and to obtain significant bounds in higher dimensions. Key contributions include a complete solution for C_1(w), a broad framework for d-stackability and d-slantability with structural results for short words, and concrete bounds for ABB-type words via a local LP method and a Fourier-based bound. The work also reveals deep connections to the modular n-queens problem, offering new perspectives on tiling-like questions and proposing several open problems for higher-dimensional regimes and longer words with potential algorithmic consequences.

Abstract

Given a word $w$, what is the maximum possible number of appearances of $w$ reading contiguously along any of the directions in $\{-1, 0, 1\}^d \setminus \{\mathbf{0}\}$ in a large $d$-dimensional grid (as in a word search)? Patchell and Spiro first posed a version of this question, which Alon and Kravitz completely answered for a large class of "well-behaved" words, including those with no repeated letters. We study the general case, which exhibits greater variety and is often more complicated (even for $d=1$). We also discuss some connections to other problems in combinatorics, including the storied $n$-queens problem.

Words with Repeated Letters in a Grid

TL;DR

The paper extends the study of word occurrences in large d-dimensional grids beyond distinct-letter words to those with repeated letters, introducing C_d(w) and the concept of w-extremal grids. It develops a multi-faceted methodology combining additive formulations, combinatorial reductions, local linear programming bounds, and Fourier analysis to establish sharp results in one and two dimensions and to obtain significant bounds in higher dimensions. Key contributions include a complete solution for C_1(w), a broad framework for d-stackability and d-slantability with structural results for short words, and concrete bounds for ABB-type words via a local LP method and a Fourier-based bound. The work also reveals deep connections to the modular n-queens problem, offering new perspectives on tiling-like questions and proposing several open problems for higher-dimensional regimes and longer words with potential algorithmic consequences.

Abstract

Given a word , what is the maximum possible number of appearances of reading contiguously along any of the directions in in a large -dimensional grid (as in a word search)? Patchell and Spiro first posed a version of this question, which Alon and Kravitz completely answered for a large class of "well-behaved" words, including those with no repeated letters. We study the general case, which exhibits greater variety and is often more complicated (even for ). We also discuss some connections to other problems in combinatorics, including the storied -queens problem.

Paper Structure

This paper contains 36 sections, 32 theorems, 97 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Introductory results
  3. Main results
  4. Definitions, techniques, and organization
  5. Definitions and notation
  6. Higher dimensions from one dimension
  7. Additive formulation
  8. Combinatorial reductions
  9. Local analysis and linear programming
  10. Fourier analysis
  11. Organization
  12. One dimension
  13. Examples and constructions
  14. Lower bound
  15. Upper bound
  16. ...and 21 more sections

Key Result

Theorem 1.2

Let $w$ be any word. There exists a closed formWe state this closed form in prop:1d and the classification of extremal grids in prop:1d-structure. for $C_1(w)$ and a simple classification of all extremal grids.

Figures (12)

  • Figure 1: The weight function used implicitly in the proof of \ref{['lem:AB-elementary']} to upper-bound $C_2(\mathsf{AB})$.
  • Figure 2: The weight function used to upper-bound $C_2(\mathsf{ABB})$.
  • Figure 3: The weight function used to upper-bound $C_2(\mathsf{ABCC})$.
  • Figure 4: The weight function used to upper-bound $C_2(\mathsf{BABBB})$. Note that each arrow in this figure starts at the $\mathsf{A}$ and points towards the $\mathsf{BBB}$ (and away from the single $\mathsf{B}$) in the appearance of $w$ the arrow indicates.
  • Figure 5: The grid $\Gamma$ on the left has $c_2(\mathsf{ABBB}, \Gamma) = 8/5.$ This is greater than the concentration of $\mathsf{ABBB}$ in the right grid $\Gamma'$, which is constant in all but one coordinate, and has the maximum concentration $c_2(\mathsf{ABBB}, \Gamma') = 3/2 = 3C_1(\mathsf{ABBB})$ among all grids constant in all but one coordinate.
  • ...and 7 more figures

Theorems & Definitions (69)

  • Theorem 1.2: Resolving the one-dimensional case
  • Proposition 1.3: General bounds on $C_d(w)$
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1: alonkravitz
  • Lemma 2.2
  • proof
  • proof : Proof of \ref{['prop:simple-d-bound']}
  • Proposition 2.6
  • Proposition 3.1
  • ...and 59 more