A049021 -id:A049021 - cp's OEIS frontend

A342141 The number of generic rectangulations with n rectangles.

1, 2, 6, 24, 116, 642, 3938, 26194, 186042, 1395008, 10948768, 89346128, 754062288, 6553942722, 58457558394, 533530004810, 4970471875914, 47169234466788, 455170730152340, 4459456443328824, 44300299824885392, 445703524836260400, 4536891586511660256, 46682404846719083048, 485158560873624409904, 5089092437784870584576, 53845049871942333501408

Offset: 1

Peter Kagey, Mar 01 2021

nonn

a(n) is also the number of two-clumped permutations of size n. Two-clumped permutations are (3-51-2-4, 3-51-4-2, 2-4-51-3, 4-2-51-3)-avoiding permutations. This class was introduced in the paper by Reading, where a bijection to generic rectangulations was also given. - Andrei Asinowski, Sep 02 2024

Andrei Asinowski, Jean Cardinal, Stefan Felsner, and Éric Fusy, Combinatorics of rectangulations: Old and new bijections, arXiv:2402.01483 [math.CO], 2023. See p. 11 and p. 27.
Jean Cardinal and Vincent Pilaud, Rectangulotopes, arXiv:2404.17349 [math.CO], 2024. See p. 18.
CombOS - Combinatorial Object Server, Generate generic rectangulations
Éric Fusy, Erkan Narmanli, and Gilles Schaeffer, On the enumeration of plane bipolar posets and transversal structures, arXiv:2105.06955 [math.CO], 2021-2023. See p. 16.
Arturo Merino and Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, arXiv:2103.09333 [math.CO], 2021.
Nathan Reading, Generic rectangulations, arXiv:1105.3093 [math.CO], 2011-2012.

Cf. A049021, A340984.

A340984 Number of prime rectangle tilings with n tiles up to equivalence.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 2, 6, 29, 119, 600

Offset: 1

Drake Thomas, Feb 01 2021

Say that a tiling of a rectangle by other rectangles is prime if the only sub-rectangles in the tiling are those formed by a single tile. Say that two tilings are equivalent if there exists an inclusion/overlap-preserving bijection between the vertices, edges, and faces of every rectangle in them.
Problem 69 in Hugo Steinhaus's One Hundred Problems In Elementary Mathematics asks the reader to show that a(3) = a(4) = 0, and that there exist prime dissections for 5, 7, and 8 in which the pieces are of equal area. It cites Czesław Ryll-Nardzewski as proving that a(6) = 0, though this is not difficult to show by hand. The book also provides diagrams of both n = 7 solutions and four of the six n = 8 solutions.
Chung et al.'s paper Tiling Rectangles with Rectangles shows that the sequence grows at least as fast as c*2^(n/7) for some positive constant c, and states without proof that it is bounded above by 20000^n.

			For n = 5 the a(5) = 1 example looks like
   _____
  | |___|
  |_|_| |
  |___|_|
.
For n = 7 the a(7) = 2 examples look like
   _______    _______
  | |_____|  |_____| |
  |_|___| |  |___| | |
  |   |_|_|  | |_|_|_|
  |___|___|  |_|_____|

F. R. K. Chung, E. N. Gilbert, R. L. Graham, J. B. Shearer, and J. H. van Lint, Tiling Rectangles with Rectangles, Mathematics Magazine, 1982.
Math StackExchange, How many "prime" rectangle tilings are there?
Benjamin D. Prins, All tilings for n = 9, 10, 11
Reddit, Distributing a rectangular inheritance.

Cf. A049021, A053740.

a(9)-a(11) from Benjamin D. Prins, Jun 13 2025

A375129 Number of combinatorially distinct ways to dissect a rectangle into n rectangles, taking into account the ordering of the lines that extend the sides.

Original entry on oeis.org

1, 1, 2, 7, 24, 126, 815, 6465, 58072, 578663

Offset: 1

Andrey Zabolotskiy, Jul 31 2024

Dissections related by rotations and reflections are considered equivalent (unlike in A342141).

			All dissections into n=4 pieces are shown in Peter Kagey's illustration, they are the same as the ones counted by A049021.
The following two dissections (labeled "Grating (3,3), 5 fronts, 0401, C_2" and "Grating (2,3), 5 fronts, 0401, K_4" in Bloch's catalog) into n=5 pieces
  (1) ┌─┬─┬─┐   (2) ┌─┬─┬─┐
      ├─┤ │ │       ├─┤ ├─┤
      │ │ ├─┤       └─┴─┴─┘
      └─┴─┴─┘
  are considered distinct by this sequence and by A375131, because the lines extending the inner horizontal sides go in the different order:
  (1) ┌─┬─┬─┐   (2) ┌─┬─┬─┐
      A─B │ │       A─B C─D
      │ │ C─D       └─┴─┴─┘
      └─┴─┴─┘
  in dissection (1), the line AB is above line CD, while in dissection (2) AB and CD is the same line. (One could also slide the side AB below CD, but this sequence would not distinguish that new dissection from (1) because it would be equivalent to the mirror image of (1).) However, A049021 views these two dissections as equivalent. A375130 and A375132 distinguish between these dissections but do not include dissection (2) at all because it has an "alignment": two internal sides AB and CD, even though they are not connected through a 4-way junction (or a sequence of sides with the same orientation, connected through 4-way junctions), still extend to coinciding lines.

J. P. Steadman, Architectural Morphology, Pion Limited, 1983. See Table 5.2 on p. 59.

C. J. Bloch, Catalogue of small rectangular plans, Environment and Planning B, 6 (1979), 155-190.
C. J. Bloch and R. Krishnamurti, The Counting of Rectangular Dissections, Environ. Plann. B, 5 (1978), 207-214.
Peter Kagey, Example of the a(4)=7 dissections into n=4 pieces.

Cf. A049021, A340984, A342141, A375130, A375131, A375132.

A375130 Number of nonaligned dissections of a rectangle into n rectangles.

Original entry on oeis.org

1, 1, 2, 7, 23, 119, 735, 5527, 46204, 423724

Offset: 1

Andrey Zabolotskiy, Jul 31 2024

Among the dissections counted by A375129, this sequence counts only those without "alignments". See A375129 for the details.

J. P. Steadman, Architectural Morphology, Pion Limited, 1983. See Table 5.2 on p. 59.

Cf. A049021, A340984, A342141, A375129, A375131, A375132.

A375131 Number of trivalent dissections of a rectangle into n rectangles.

Original entry on oeis.org

1, 1, 2, 6, 22, 108, 668, 5026, 43005, 389803

Offset: 1

Andrey Zabolotskiy, Jul 31 2024

Among the dissections counted by A375129, this sequence counts only those without 4-way junctions. See A375129 for the details.

J. P. Steadman, Architectural Morphology, Pion Limited, 1983. See Table 5.2 on p. 59.

Cf. A049021, A340984, A342141, A375129, A375130, A375132.

A375132 Number of nonaligned trivalent dissections of a rectangle into n rectangles.

Original entry on oeis.org

1, 1, 2, 6, 21, 101, 591, 4168, 32754, 282605

Offset: 1

Andrey Zabolotskiy, Jul 31 2024

J. P. Steadman, Architectural Morphology, Pion Limited, 1983. See Table 5.2 on p. 59.

Cf. A049021, A340984, A342141, A375129, A375130, A375131.

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A342141 The number of generic rectangulations with n rectangles.

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A340984 Number of prime rectangle tilings with n tiles up to equivalence.

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A375129 Number of combinatorially distinct ways to dissect a rectangle into n rectangles, taking into account the ordering of the lines that extend the sides.

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A375130 Number of nonaligned dissections of a rectangle into n rectangles.

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A375131 Number of trivalent dissections of a rectangle into n rectangles.

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A375132 Number of nonaligned trivalent dissections of a rectangle into n rectangles.

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