### Abstract

A dyadic interval is an interval of the form [j/2
^{k},(j+ 1)/2
^{k}], where j and k are integers, and a dyadic rectangle is a rectangle with sides parallel to the axes whose projections on the axes are dyadic intervals. Let u
_{n} count the number of ways of partitioning the unit square into 2
^{n} dyadic rectangles, each of area 2
^{-n}. One has u
_{0} = 1, u
_{1} = 2 and u
_{n} = 2u
_{n-1}
^{2} - u
_{n-2}
^{4}. This paper determines an asymptotic formula for a solution to this nonlinear recurrence for generic real initial conditions. For almost all real initial conditions there are real constants ω and β (depending on u
_{0},u
_{1}) with ω > 0 such that for all sufficiently large n one has the exact formula u
_{n} = ω
^{2n}g(βλ
^{n}), where λ = 2√5 - 4 ≈ 0.472, and g(z) = ∑
_{j=0}
^{∞} c
_{j}z
^{j}, in which c
_{0} = (-1 + √5)/2, c
_{1} = (2 - √5)/2, all coefficients c
_{j} lie in the field (√5), and the power series converges for |z| < 0.16. These results apply to the initial conditions u
_{0} = 1, u
_{1} = 2 with ω ≈ 1.845 and β ≈ 0.480. The exact formula for u
_{n} then holds for all n ≥ 2. The proofs are based on an analysis of the holomorphic dynamics of iterating the rational function R(z) = 2 - 1/z
^{2}. Keywords: Asymptotic enumeration; Holomorphic dynamic PII: S0012-365X(02)00508-3.

Original language | English (US) |
---|---|

Pages (from-to) | 481-499 |

Number of pages | 19 |

Journal | Discrete Mathematics |

Volume | 257 |

Issue number | 2-3 |

State | Published - Nov 28 2002 |

### Fingerprint

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*257*(2-3), 481-499.

**Counting dyadic equipartitions of the unit square.** / Lagarias, Jeffrey C.; Spencer, Joel H.; Vinson, Jade P.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 257, no. 2-3, pp. 481-499.

}

TY - JOUR

T1 - Counting dyadic equipartitions of the unit square

AU - Lagarias, Jeffrey C.

AU - Spencer, Joel H.

AU - Vinson, Jade P.

PY - 2002/11/28

Y1 - 2002/11/28

N2 - A dyadic interval is an interval of the form [j/2 k,(j+ 1)/2 k], where j and k are integers, and a dyadic rectangle is a rectangle with sides parallel to the axes whose projections on the axes are dyadic intervals. Let u n count the number of ways of partitioning the unit square into 2 n dyadic rectangles, each of area 2 -n. One has u 0 = 1, u 1 = 2 and u n = 2u n-1 2 - u n-2 4. This paper determines an asymptotic formula for a solution to this nonlinear recurrence for generic real initial conditions. For almost all real initial conditions there are real constants ω and β (depending on u 0,u 1) with ω > 0 such that for all sufficiently large n one has the exact formula u n = ω 2ng(βλ n), where λ = 2√5 - 4 ≈ 0.472, and g(z) = ∑ j=0 ∞ c jz j, in which c 0 = (-1 + √5)/2, c 1 = (2 - √5)/2, all coefficients c j lie in the field (√5), and the power series converges for |z| < 0.16. These results apply to the initial conditions u 0 = 1, u 1 = 2 with ω ≈ 1.845 and β ≈ 0.480. The exact formula for u n then holds for all n ≥ 2. The proofs are based on an analysis of the holomorphic dynamics of iterating the rational function R(z) = 2 - 1/z 2. Keywords: Asymptotic enumeration; Holomorphic dynamic PII: S0012-365X(02)00508-3.

AB - A dyadic interval is an interval of the form [j/2 k,(j+ 1)/2 k], where j and k are integers, and a dyadic rectangle is a rectangle with sides parallel to the axes whose projections on the axes are dyadic intervals. Let u n count the number of ways of partitioning the unit square into 2 n dyadic rectangles, each of area 2 -n. One has u 0 = 1, u 1 = 2 and u n = 2u n-1 2 - u n-2 4. This paper determines an asymptotic formula for a solution to this nonlinear recurrence for generic real initial conditions. For almost all real initial conditions there are real constants ω and β (depending on u 0,u 1) with ω > 0 such that for all sufficiently large n one has the exact formula u n = ω 2ng(βλ n), where λ = 2√5 - 4 ≈ 0.472, and g(z) = ∑ j=0 ∞ c jz j, in which c 0 = (-1 + √5)/2, c 1 = (2 - √5)/2, all coefficients c j lie in the field (√5), and the power series converges for |z| < 0.16. These results apply to the initial conditions u 0 = 1, u 1 = 2 with ω ≈ 1.845 and β ≈ 0.480. The exact formula for u n then holds for all n ≥ 2. The proofs are based on an analysis of the holomorphic dynamics of iterating the rational function R(z) = 2 - 1/z 2. Keywords: Asymptotic enumeration; Holomorphic dynamic PII: S0012-365X(02)00508-3.

UR - http://www.scopus.com/inward/record.url?scp=33751006768&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33751006768&partnerID=8YFLogxK

M3 - Article

VL - 257

SP - 481

EP - 499

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 2-3

ER -