TY - JOUR

T1 - Counting dissections into integral squares

AU - Oh, Seungsang

N1 - Funding Information:
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government ( MSIP ) (No. NRF-2017R1A2B2007216 ).
Publisher Copyright:
© 2022 Elsevier B.V.

PY - 2022/5

Y1 - 2022/5

N2 - 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.

AB - 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.

KW - Dissection

KW - Rectangled rectangle

KW - Squared rectangle

KW - Tiling

UR - http://www.scopus.com/inward/record.url?scp=85123122591&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2022.112803

DO - 10.1016/j.disc.2022.112803

M3 - Article

AN - SCOPUS:85123122591

SN - 0012-365X

VL - 345

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 5

M1 - 112803

ER -