Automatic Enumeration of Tilings by Polyominoes

• Based on this paper by Lucy Martinez

Usage

Example using TILINGSapps.txt.

• Save both files: TILINGSapps.txt and TILINGSapps.txt in the same folder. Use TILINGSapps.txt and stay in the same directory. Get into Mape and type:

  read `TILINGSapps.txt:`

• The following will be prompted

                First Written: Jan. 2006: tested for Maple 10
                   Version of Jan. , 2006:

                This is TILINGS, A Maple package
accompanying Shalsoh B. Ekhad and Doron Zeilberger's  article: 
                    "Automatic CounTilings" 

                -------------------------------
        Version of Sept. 30, 2015, procedure Dimer added
                -------------------------------

        The most current version is available on WWW at:
    http://www.math.rutgers.edu/~zeilberg/tokhniot/TILINGS .
Please report all bugs to: zeilberg at math dot rutgers dot edu .

      For general help, and a list of the MAIN functions,
 type "ezra();". For specific help type "ezra(procedure_name);" 
     For a list of the supporting functions type: ezra1();
   ----------------------------------------------------------

NOTE: The above script is for a Maple package separate from the one below (see note below).

   ----------------------------------------------------------

  This is TILINGSapps.txt, a Maple package, accompanying the article:
        Automatic Enumeration of Tilings by Polyominoes
                       by Lucy Martinez.
        This Maple package, as well as the article, are 
                        available from:
            https://github.com/marti310/TILINGSapps

NOTE: This package and the accompanying paper are applications of the 2006 paper written by
    Shalosh B. Ekhad and Doron Zeilberger titled Automatic CounTilings.

      Please report bugs to lucyadrianamartinez@gmail.com 

                 -----------------------------

    For a list of the procedures type Help(), for help with
        a specific procedure, type Help(procedure_name)

                 -----------------------------
                 -----------------------------


For a list of the supporting procedures type Help1(), for help with
        a specific procedure, type Help(procedure_name)

                 -----------------------------
                 -----------------------------


 For a list of the Paper procedures type HelpP(), for help with

        a specific procedure, type Help(procedure_name)

                 -----------------------------

• Inputting GFrot(4,{[0,0],[1,0],[0,1],[0,2]},t); , outputs the generating function from the recent American Mathematical Monthly paper (found here), which is the generating function for the number of tilings of a 2n by 4 rectangle using L-tetrominoes, where only rotations are allowed.

  • Input:

    GFrot(4,{[0,0],[1,0],[0,1],[0,2]},t);
    

  • Output:

    -(t^4 - 1)/(t^8 + t^6 - 3*t^4 - t^2 + 1)
    

• If you want to see a short article containing theorems that, for all rotations of the L-polyomino with <=r cells (4<=r<=9) and every rectangular board of width k (1<=k<=9), provide:

  • The generating function for tilings of a k by n rectangle by all four rotations of the L-polyomino.
  • The asymptotics for these sequences.
  • The first 30 terms of the corresponding counting sequences.

  Then, see

  • Input file: PaperLpolyomino_Input.txt click here.
  • Output file: PaperLpolyomino.txt Click here.

• If you want to see a short article containing theorems that, for all rotations of the T-tetromino and every rectangular board of width k (1<=k<=16 and k is a multiple of 4), provide:

  • The generating function for tilings of a k by 4n rectangle by all four rotations of the T-tetromino.
  • The asymptotics for these sequences.
  • The first 30 terms of the corresponding counting sequences.

  Then, see

  • Input file: PaperTtetromino_Input.txt click here.
  • Output file: PaperTtetromino.txt Click here.

• If you want to see a short article containing theorems that, for every free polyomino with r cells (1 <= r <= 4) and every rectangular board of width k (1 <= k <= 4), provide:

  • The generating function for tilings of a k by n rectangle by rotations of the polyomino.
  • The asymptotics for these sequences.
  • The first 30 terms of the corresponding counting sequences.

  Then, see

  • Input file: Paper1_Input.txt click here.
  • Output file: Paper1.txt Click here.

• If you want to see a short article containing theorems that, for every free polyomino with r cells (1 <= r <= 4) and every rectangular board of width k (1 <= k <= 4), provide:

  • The generating function for tilings of a k by n rectangle by all images of the polyomino (rotations and reflections).
  • The asymptotics for these sequences.
  • The first 30 terms of the corresponding counting sequences.

  Then, see

  • Input file: Paper2_Input.txt click here.
  • Output file: Paper2.txt click here.

• If you want to see a short article containing theorems that, for all fixed polyominoes with r cells (1 <= r <= 4) and every rectangular board of width k (1 <= k <= 4), provide:

  • The generating function for tilings of a k by n rectangle by all the fixed polyominoes with exactly r cells.
  • The asymptotics for these sequences.
  • The first 30 terms of the corresponding counting sequences.

  Then, see

  • Input file: Paper3_Input.txt click here.
  • Output file: Paper3.txt click here.

• If you want to see a short article containing theorems that, for for all fixed polyominoes with r cells (1 <= r <= 4) and every rectangular board of width k (1 <= k <= 4), provide:

  • The generating function for tilings of a k by n rectangle by the union of all the fixed polyominoes with up to r cells.
  • The asymptotics for these sequences.
  • The first 30 terms of the corresponding counting sequences.

  Then, see

  • Input file: Paper4_Input.txt click here.
  • Output file: Paper4.txt click here.