Written: Dec. 2023

We study maximal seating arrangments, either on a line, or in a rectangular auditorium with a fixed number of columns but an arbitrary number of rows, that obey any prescribed set of `social distancing' restrictions. In addition to enumeration, we study the statistical distribution of the density, and give efficient algoirthms for generating these at random.

Maple packages

• SD1row.txt, a Maple package to investigate 1D social distancing with varios restrictions.

• Houses.txt, a Maple package to investigate the original T-avoding situation in the Puljiz-Sebek-Zubrinic Amer. Mathematical Monthy article v. 130 No. 10 (Dec. 2023), 915-928.

Sample Input and Output for SD1row.txt

• If you want so see generating functions and RSA simulations for maximal configurations of seatings in one row avoding a contiguous block of b people for b from 2 to 6
  the input file yields the output file.

Sample Input and Output for Houses.txt

• If you want to see exact information about enumerating MAXIMAL rectangular housing developments with THREE columns, avoiding T (i.e. that do not block the sunshine), and comparing their statistics with the RSA (Random Sequential Adsorption, NOT Rivest-Shamir-Adleman) approach by simulating 10000 times for 100x3 0-1 matrices
  the input file yields the output file.

  
  If you do the same but simulating 100000 times then
  the input file yields the output file.

  ---

• If you want to see exact information about enumerating MAXIMAL rectangular housing developments with FOUR columns, avoiding T (i.e. that do not block the sunshine), and comparing their statistics with the RSA (Random Sequential Adsorption, NOT Rivest-Shamir-Adleman) approach by simulating 10000 times for 100x4 0-1 matrices
  the input file yields the output file.

  
  If you do the same but simulating 100000 times then
  the input file yields the output file.

  ---

• If you want to see exact information about enumerating MAXIMAL rectangular housing developments with FIVE columns, avoiding T (i.e. that do not block the sunshine), and comparing their statistics with the RSA (Random Sequential Adsorption, NOT Rivest-Shamir-Adleman) approach by simulating 10000 times for 100x5 0-1 matrices
  the input file yields the output file.

  
  If you do the same but simulating 100000 times then
  the input file yields the output file.

  ---

• If you want to see exact information about enumerating MAXIMAL rectangular housing developments with SIX columns, avoiding T (i.e. that do not block the sunshine), and comparing their statistics with the RSA (Random Sequential Adsorption, NOT Rivest-Shamir-Adleman) approach by simulating 10000 times for 100x6 0-1 matrices
  the input file yields the output file.

  
  If you do the same but simulating 100000 times then
  the input file yields the output file.

  ---

  ---

• If you want to see exact information about enumerating 0-1 matrices with THREE columns, avoiding without neighboring ones (both horizontally and vertically, i.e. avoiding dimers) and comparing their statistics with the RSA (Random Sequential Adsorption, NOT Rivest-Shamir-Adleman) approach by simulating 1000 times for 100x3 0-1 matrices
  the input file yields the output file.

• If you want to see exact information about enumerating 0-1 matrices with FOUR columns, avoiding without neighboring ones (both horizontally and vertically, i.e. avoiding dimers) and comparing their statistics with the RSA (Random Sequential Adsorption, NOT Rivest-Shamir-Adleman) approach by simulating 1000 times for 100x4 0-1 matrices
  the input file yields the output file.

• If you want to see exact information about enumerating 0-1 matrices with FIVE columns, avoiding without neighboring ones (both horizontally and vertically, i.e. avoiding dimers) and comparing their statistics with the RSA (Random Sequential Adsorption, NOT Rivest-Shamir-Adleman) approach by simulating 1000 times for 100x5 0-1 matrices
  the input file yields the output file.

  ---

  ---

• If you want to see exact information about enumerating 0-1 matrices with THREE columns, avoiding without neighboring ones (both horizontally and vertically and diagonally, i.e. non-attacking kings) and comparing their statistics with the RSA (Random Sequential Adsorption, NOT Rivest-Shamir-Adleman) approach by simulating 1000 times for 100x3 0-1 matrices
  the input file yields the output file.

• If you want to see exact information about enumerating 0-1 matrices with FOUR columns, avoiding without neighboring ones (both horizontally and vertically and diagonally, i.e. non-attacking kings) and comparing their statistics with the RSA (Random Sequential Adsorption, NOT Rivest-Shamir-Adleman) approach by simulating 1000 times for 100x4 0-1 matrices
  the input file yields the output file.

• If you want to see exact information about enumerating 0-1 matrices with FIVE columns, avoiding without neighboring ones (both horizontally and vertically and diagonally, i.e. non-attacking kings) and comparing their statistics with the RSA (Random Sequential Adsorption, NOT Rivest-Shamir-Adleman) approach by simulating 1000 times for 100x5 0-1 matrices
  the input file yields the output file.

  ---

  ---

• If you want to see exact information about enumerating 0-1 matrices with THREE columns, avoiding a 2x2 square of ones,i.e. the pattern
  11
  11
  and comparing their statistics with the RSA (Random Sequential Adsorption, NOT Rivest-Shamir-Adleman) approach by simulating 1000 times for 100x5 0-1 matrices
  the input file yields the output file.

• If you want to see exact information about enumerating 0-1 matrices with FOUR columns, avoiding a 2x2 square of ones,i.e. the pattern
  11
  11
  and comparing their statistics with the RSA (Random Sequential Adsorption, NOT Rivest-Shamir-Adleman) approach by simulating 1000 times for 100x5 0-1 matrices
  the input file yields the output file.

• If you want to see exact information about enumerating 0-1 matrices with FIVE columns, avoiding a 2x2 square of ones,i.e. the pattern
  11
  11
  and comparing their statistics with the RSA (Random Sequential Adsorption, NOT Rivest-Shamir-Adleman) approach by simulating 1000 times for 100x5 0-1 matrices
  the input file yields the output file.

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