Counting dissections into integral squares

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A squared rectangle is a rectangle dissected into squares. Similarly a rectangled rectangle is a rectangle dissected into rectangles. The classic paper ‘The dissection of rectangles into squares’ of Brooks, Smith, Stone and Tutte described a beautiful connection between squared rectangles and harmonic functions. In this paper we count dissections of a rectangle into a set of integral squares or a set of integral rectangles. Here, some squares and rectangles may have the same size. We introduce a method involving a recurrence relation of large sized matrices to enumerate squared and rectangled rectangles of a given sized rectangle and propose the asymptotic behavior of their growth rates.

Original languageEnglish
Article number112803
JournalDiscrete Mathematics
Issue number5
Publication statusPublished - 2022 May


  • Dissection
  • Rectangled rectangle
  • Squared rectangle
  • Tiling

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


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