Counting dyadic equipartitions of the unit square

Jeffrey C. Lagarias, Joel H. Spencer, Jade P. Vinson

Research output: Contribution to journalArticle

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 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.

Original languageEnglish (US)
Pages (from-to)481-499
Number of pages19
JournalDiscrete Mathematics
Volume257
Issue number2-3
StatePublished - Nov 28 2002

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Equipartition
Holomorphic Dynamics
Rectangle
Counting
Initial conditions
Interval
Rational functions
Unit
Asymptotic Enumeration
Asymptotic Formula
Power series
Rational function
Recurrence
Partitioning
Count
Projection
Converge
Integer
Coefficient

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

Lagarias, J. C., Spencer, J. H., & Vinson, J. P. (2002). Counting dyadic equipartitions of the unit square. Discrete Mathematics, 257(2-3), 481-499.

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

In: Discrete Mathematics, Vol. 257, No. 2-3, 28.11.2002, p. 481-499.

Research output: Contribution to journalArticle

Lagarias, JC, Spencer, JH & Vinson, JP 2002, 'Counting dyadic equipartitions of the unit square', Discrete Mathematics, vol. 257, no. 2-3, pp. 481-499.
Lagarias JC, Spencer JH, Vinson JP. Counting dyadic equipartitions of the unit square. Discrete Mathematics. 2002 Nov 28;257(2-3):481-499.
Lagarias, Jeffrey C. ; Spencer, Joel H. ; Vinson, Jade P. / Counting dyadic equipartitions of the unit square. In: Discrete Mathematics. 2002 ; Vol. 257, No. 2-3. pp. 481-499.
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