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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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\textit{M. Abrahamsen} and \textit{S. Eilers}, Exp. Math. 20, No. 2, 145--152 (2011; Zbl 1262.05006)

### 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) |

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