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On the asymptotic enumeration of LEGO structures. (English) Zbl 1262.05006

Summary: We investigate experimentally the growth regimes of the number of LEGO structures that can be constructed contiguously from \(n\) blocks of equal shape and color.

MSC:

05A16 Asymptotic enumeration
05-04 Software, source code, etc. for problems pertaining to combinatorics
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Full Text: DOI Euclid

Online Encyclopedia of Integer Sequences:

Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 2 X 4.
Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 1 X 2.
Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 1 X 2 which is symmetric after a rotation by 180 degrees.
Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 1 X 2 which is flat, i.e., with all blocks in parallel position.
Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 1 X 2 which is flat, i.e., with all blocks in parallel position and symmetric after a rotation by 180 degrees.
Number of ways to build a contiguous building with n LEGO blocks of size 1 X 2 on top of a fixed block of the same size.
Number of ways to build a contiguous building with n LEGO blocks of size 1 X 2 on top of a fixed block of the same size so that the building is symmetric after a rotation by 180 degrees.
Number of ways to build a contiguous building with n LEGO blocks of size 1 X 2 on top of a fixed block of the same size so that the building is flat, i.e., with all blocks in parallel position.
Number of ways to build a contiguous building with n LEGO blocks of size 1 X 2 on top of a fixed block of the same size so that the building is flat, i.e., with all blocks in parallel position and symmetric after a rotation by 180 degrees.
Number of ways to build a contiguous building with n LEGO blocks of size 1 X 6 on top of a fixed block of the same size so that the building is flat, i.e., with all blocks in parallel position and symmetric after a rotation by 180 degrees.
Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 3 X 3.
Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 3 X 3 which is symmetric after a rotation by 180 degrees.
Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 3 X 3 which is symmetric after a rotation by 90 degrees.
Number of ways to build a contiguous building with n LEGO blocks of size 3 X 3 on top of a fixed block of the same size.
Number of ways to build a contiguous building with n LEGO blocks of size 3 X 3 on top of a fixed block of the same size so that the building is symmetric after a rotation by 180 degrees.
Number of ways to build a contiguous building with n LEGO blocks of size 4 X 4 on top of a fixed block of the same size so that the building is symmetric after a rotation by 180 degrees.
Number of ways to build a contiguous building with n LEGO blocks of size 4 X 4 on top of a fixed block of the same size so that the building is symmetric after a rotation by 90 degrees.
Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 5x5.
Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 5x5 which is symmetric after a rotation by 180 degrees.
Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 5x5 which is symmetric after a rotation by 90 degrees.
Number of ways to build a contiguous building with n LEGO blocks of size 5x5 on top of a fixed block of the same size.
Number of ways to build a contiguous building with n LEGO blocks of size 5 X 5 on top of a fixed block of the same size so that the building is symmetric after a rotation by 180 degrees.
Number of ways to build a contiguous building with n LEGO blocks of size 5 X 5 on top of a fixed block of the same size so that the building is symmetric after a rotation by 90 degrees.

References:

[1] Flajolet [Flajolet and Sedgewick 08] P., Analytic Combinatorics (2008)
[2] DOI: 10.1002/andp.18471481202 · doi:10.1002/andp.18471481202
[3] DOI: 10.1007/BF02183684 · Zbl 1080.82541 · doi:10.1007/BF02183684
[4] Matousek [Matousek 98] J., Invitation to Discrete Mathematics (1998)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.