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Patterns in sequences

Richard Kenyon

TL;DR

The paper develops a rigorous framework for understanding pattern densities in binary sequences by introducing sublebesgue measures as limit objects and a variational principle that maximizes entropy under pattern-density constraints. It provides explicit feasibility regions for two-pattern cases, derives the structure of entropy-maximizing limit shapes (typically yielding $H'(y)=\frac{1}{1-e^{p(y)}}$ with $p$ a low-/moderate-degree polynomial), and analyzes concrete families such as $0,10,110,\dots,1^k0$ and the four-pattern $1010$. Through connections to the Heisenberg group and total positivity, the work links discrete pattern counting to continuous optimization and matrix-positivity phenomena, offering both exact results (e.g., the $1010$ case with maximum $\rho_{1010}=\frac{3}{4e^2}$ at $\rho_1=1/2$) and a general variational methodology. The results illuminate the typical structure of long binary sequences with prescribed pattern densities and open pathways to broader alphabets and algebraic questions about independence of pattern counts.

Abstract

We study pattern densities in binary sequences, finding optimal limit sequences with fixed pattern densities.

Patterns in sequences

TL;DR

The paper develops a rigorous framework for understanding pattern densities in binary sequences by introducing sublebesgue measures as limit objects and a variational principle that maximizes entropy under pattern-density constraints. It provides explicit feasibility regions for two-pattern cases, derives the structure of entropy-maximizing limit shapes (typically yielding with a low-/moderate-degree polynomial), and analyzes concrete families such as and the four-pattern . Through connections to the Heisenberg group and total positivity, the work links discrete pattern counting to continuous optimization and matrix-positivity phenomena, offering both exact results (e.g., the case with maximum at ) and a general variational methodology. The results illuminate the typical structure of long binary sequences with prescribed pattern densities and open pathways to broader alphabets and algebraic questions about independence of pattern counts.

Abstract

We study pattern densities in binary sequences, finding optimal limit sequences with fixed pattern densities.
Paper Structure (13 sections, 5 theorems, 65 equations, 7 figures)

This paper contains 13 sections, 5 theorems, 65 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Pattern densities and sublebesgue measures
  3. Algebraic relations
  4. Feasibility regions
  5. Variational principle
  6. $0, 10, 110,\dots,1^k0$ patterns
  7. Upper-triangular matrices
  8. Open questions
  9. Appendix
  10. $1^k0^l1^m$
  11. $11010$
  12. $10110$
  13. $10101$

Key Result

Theorem 2.1

Two sublebesgue measures are a.e. equal if and only if they have the same pattern densities. More generally, a sequence of sublebesgue measures converges in metric $d_W$ if and only if its set of pattern densities converge.

Figures (7)

  • Figure 1: A binary word can be presented as the boundary of a Young diagram: a NE lattice path where $0$ is a step east and $1$ a step north. Then $N_{10}$ is the area under the Young diagram. Here $N_{10}(0100101)=4$.
  • Figure 2: Conjecturally optimal arrangement of a $52$ card deck for the BRBR game. Starting from the origin, a north step is a black card; an east step is a red card. This arrangement gives $p\approx0.114$. Compare with Figure \ref{['F1010YD']}.
  • Figure 3: Feasibility region for patterns $1$ and $10$.
  • Figure 4: Plot of density $f(x)$ of the $\rho_{1010}$-maximizer when $\rho_1=\frac{1}{2}$ and when $\rho_1=\frac{1}{4}$.
  • Figure 5: Plot of the $\rho_{1010}$-maximizer as a NE path, when $\rho_1=\frac{1}{2}$.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Theorem 2.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 6.1
  • proof
  • Lemma 6.2
  • proof