A056814 -id:A056814 - cp's OEIS frontend

A049021 Number of topologically distinct ways to dissect a rectangle into n rectangles.

1, 1, 2, 7, 23, 116, 683, 4866

Offset: 1

The data are from Mitchell, Steadman & Liggett (MSL). Combes gives terms 1, 2, 7, 23, 116, 685, 5124. Stesney reconstructed MSL's algorithm and got 2, 7, 23, 116, 685, 4899. For some higher n, MSL's algorithm is known not to be exhaustive [Steadman, pp. 39-40]. - Andrey Zabolotskiy, Sep 26 2023

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.

Cf. A056814, A340984, A342141, A375129, A375130, A375131, A375132.

A053740 Number of prime triangle partitions of order n.

Original entry on oeis.org

1, 1, 3, 8, 62, 535, 4213

Offset: 2

N. J. A. Sloane, Sep 01 2000

A triangle partition is prime if it does not contain a triangle partition of lower order.

The order of a triangle partition is the number of smaller triangles into which the initial triangle is divided. The sequence counts only topologically distinct partitions. - M. F. Hasler, Feb 14 2024

			From _M. F. Hasler_, Feb 14 2024: (Start)
a(2) = 1 because a triangle can be divided into two smaller triangles in only one way, up to topological equivalence, namely by a straight line going through one of the vertices and a point on the opposite side.
a(3) = 1 counts the dissection of a triangle ABC into three smaller ones by three segments AD, BD, CD, where D is a point inside ABC. There are three other topologically inequivalent partitions of order 3, each using two segments, as follows: {AE, AF}, {AE, EG} and {AE, BH}, where E and F are two distinct points on BC, G is a point on AB, and H is a point on AE. It is easy to see that these aren't prime since removing the smaller triangle that has side AC leaves a triangle partition of order 2. (End)

Ed Pegg Jr., Triangles
Miroslav Vicher, Triangle Partitions
Eric Weisstein's World of Mathematics, Triangle Dissection

Cf. A056814.

cp's OEIS Frontend

A049021 Number of topologically distinct ways to dissect a rectangle into n rectangles.

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