NDLI: On the Number of Tilings of a Square by Rectangles

Content Provider	Springer Nature Link
Author	Conant, Jim Michaels, Tim
Copyright Year	2013
Abstract	We develop a recursive formula for counting the number of rectangulations of a square, i.e., the number of combinatorially distinct tilings of a square by rectangles. Our formula specializes to give a formula counting generic rectangulations, as analyzed by Reading in [5]. Our computations agree with [5] as far as was calculated and extend to the non-generic case. An interesting feature of the number of rectangulations is that it appears to have an 8-fold periodicity modulo 2. We verify this periodicity for small values of n, but the general result remains elusive, perhaps hinting at some unseen structure on the space of rectangulations, analogous to Reading’s discovery that generic rectangulations are in 1–1 correspondence with a certain class of permutations. Finally, we use discrete Morse theory to show that the space of tilings by ≤ n rectangles is homotopy-equivalent to a wedge of some number of (n−1)-dimensional spheres. Combining this result with the formulae for the number of tilings, the exact homotopy type is computed for n ≤ 28.
Ending Page	34
Page Count	14
Starting Page	21
File Format	PDF
ISSN	02180006
e-ISSN	02193094
Journal	Annals of Combinatorics
Issue Number	1
Volume Number	18
Language	English
Publisher	Springer Basel
Publisher Date	2013-11-23
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Exact enumeration problems, generating functions rectangular tilings rectangulations Classification of homotopy type Combinatorics
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics

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