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.
1, 1, 2, 7, 24, 126, 815, 6465, 58072, 578663
Offset: 1
Examples
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.
References
- J. P. Steadman, Architectural Morphology, Pion Limited, 1983. See Table 5.2 on p. 59.
Links
- 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.
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