A049021
Number of topologically distinct ways to dissect a rectangle into n rectangles.
Original entry on oeis.org
1, 1, 2, 7, 23, 116, 683, 4866
Offset: 1
- E. J. Sauda, Dissection generating algorithm (University of Louisiana), 1976.
- J. P. Steadman, Architectural Morphology, Pion Limited, London 1983, ISBN 0 85086 08605.
- C. J. Bloch, Catalogue of small rectangular plans, Environment and Planning B, 6 (1979), 155-190. [Note: this paper is related to a similar but different sequence, see A375129.]
- C. J. Bloch and R. Krishnamurti, The Counting of Rectangular Dissections, Environ. Plann. B, 5 (1978), 207-214. [Note: this paper is related to a similar but different sequence, see A375129.]
- L. Combes, Packing Rectangles into Rectangular Arrangements, Environ. Plann. B, 3 (1976), 3-32.
- Peter Kagey, Example of the a(4)=7 dissections into n=4 pieces.
- W. J. Mitchell, J. P. Steadman and R. S. Liggett, Synthesis and optimization of small rectangular floor plans, Environment and Planning B, 1976 vol. 3, 37-70.
- Michael Stesney, Rematerializing Graphs: Learning Spatial Configuration, Master's Thesis, Carnegie Mellon University, 2021.
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
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.
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
- J. P. Steadman, Architectural Morphology, Pion Limited, 1983. See Table 5.2 on p. 59.
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
- J. P. Steadman, Architectural Morphology, Pion Limited, 1983. See Table 5.2 on p. 59.
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
- J. P. Steadman, Architectural Morphology, Pion Limited, 1983. See Table 5.2 on p. 59.
A377921
Number of 3-connected Schnyder labelings with n inner faces in the quadrangular dissection, whose outer white vertices are non-isolated.
Original entry on oeis.org
0, 0, 0, 1, 0, 3, 4, 15, 42, 131, 438, 1467, 5204, 18687, 69306, 261473, 1007328, 3943155, 15679474, 63199785, 257983836, 1065186381, 4444618518, 18725820855, 79603364236, 341206137453, 1473818915658, 6411853714819, 28082121491796, 123764682835371, 548675459164202
Offset: 0
A375913
Number of strong (=generic) guillotine rectangulations with n rectangles.
Original entry on oeis.org
1, 2, 6, 24, 114, 606, 3494, 21434, 138100, 926008, 6418576, 45755516, 334117246, 2491317430, 18919957430, 146034939362, 1143606856808, 9072734766636, 72827462660824, 590852491725920, 4840436813758832, 40009072880216344, 333419662183186932, 2799687668599080296
Offset: 1
- Andrei Asinowski, Jean Cardinal, Stefan Felsner, and Éric Fusy, Combinatorics of rectangulations: Old and new bijections, arXiv:2402.01483 [math.CO], 2024, page 37.
- Arturo Merino and Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, Discrete Comput. Geom., 70(1):51-122, 2023. Page 99, Table 3, entry "12".
Cf.
A342141 (number of strong (=generic) rectangulations).
Cf.
A001181 (Baxter numbers: number of weak (=diagonal) rectangulations).
Cf.
A006318 (Schröder numbers: number of weak (=diagonal) guillotine rectangulations).
A375923
Number of permutations of size n which are both two-clumped and co-two-clumped.
Original entry on oeis.org
1, 1, 2, 6, 24, 112, 582, 3272, 19550, 122628, 800392, 5400342, 37475474, 266412680, 1934033968, 14300538652, 107471798112, 819442325086, 6329551390064, 49465665347580, 390692732060804, 3115700976866356, 25067250869113332, 203317147838575616, 1661425311693158000
Offset: 0
Cf.
A342141 (number of two-clumped permutations).
Cf.
A001181 (Baxter numbers: number of (twisted-)Baxter permutations).
Cf.
A348351 (number of permutations which are both twisted-Baxter and co-twisted-Baxter).
Showing 1-8 of 8 results.
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