# Differences

This shows you the differences between two versions of the page.

Both sides previous revision Previous revision Next revision | Previous revision | ||

table_seating [2021-06-17 07:48] – timbo | table_seating [2022-04-04 03:32] (current) – nik | ||
---|---|---|---|

Line 1: | Line 1: | ||

==== Table Seatings ==== | ==== Table Seatings ==== | ||

- | |||

The general problem is arranging a group of people into a number of tables so that everyone sits with everyone else. There are multiple versions for this. | The general problem is arranging a group of people into a number of tables so that everyone sits with everyone else. There are multiple versions for this. | ||

- | * The strict version is that all tables are the same size and that after the required number of rounds, everyone has shared a table with every other person exactly once | + | * The **strict** version is that all tables are the same size and that after the required number of rounds, everyone has shared a table with every other person exactly once |

- | * The lower version requires that each person shares a table with each other person at most once | + | * The **lower version** requires that each person shares a table with each other person at most once |

- | * The upper version requires that each person shares a table with each other person at least once | + | * The **upper version** requires that each person shares a table with each other person at least once |

Some general thoughts. Each sitting defines a partition of the set of people, each part is one table. | Some general thoughts. Each sitting defines a partition of the set of people, each part is one table. | ||

- | ===Strict=== | + | ====Strict==== |

A strict version is an affine plane. Example 25 people in 5 tables of 5, Point set is Z_5 x Z_5,we take the tables to be the lines L(a, | A strict version is an affine plane. Example 25 people in 5 tables of 5, Point set is Z_5 x Z_5,we take the tables to be the lines L(a, | ||

- | More generally we want a [[https:// | + | More generally we want a [[https://. We want resolvable (v,k,1) designs. Resolvable is the parallelism. Maybe there is something like discrete hyperbolic geometry to deal with this, but we seem to have better combinatorial ideas below. |

+ | | ||

+ | | ||

+ | Versions include [[https://. | ||

+ | For pairs, resolvable (v,2,1) 2-designs exist only for even v, v >= 4. | ||

- | Strict versions include [[https://en.wikipedia.org/wiki/Kirkman%27s_schoolgirl_problem|Kirkman']] (15 children walk in groups of 3, can they do this so that all pairs of girls walk together exactly once over a whole week) {[[https://.org/ | + | Table size 4: Resolvable (v,k,1)- 2-designs. |

+ | [[https://www.semanticscholar.org/paper/The-spectrum-of-resolvable-designs-with-block-size-Vasiga-Furino/|This paper]] says that nessesary numerical conditions are sufficient except for a case that need not concern us. | ||

- | === Lower Version=== | + | ==== Lower Version==== |

Just leave out some sittings on a strict version. Perhaps add a few nonexistant people to get a better distribution of people on tables with not all tables always full. | Just leave out some sittings on a strict version. Perhaps add a few nonexistant people to get a better distribution of people on tables with not all tables always full. | ||

- | === Upper Version === | + | ==== Upper Version ==== |

The " | The " |