Show pageOld revisionsBacklinksBack to top You've loaded an old revision of the document! If you save it, you will create a new version with this data. Media Files==== 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 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 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. ===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,b)={(x,y)| y=ax+b} and L(a)={(a,y)| y in Z_5}, the sitting is a parallel class (the 5 lines with the same slope a), so we have 6 sittings, L(a,b) for a=0,1,2,3,4 and then the parallel class of L(a). More generally we want a [[https://en.wikipedia.org/wiki/Block_design#Resolvable_2-designs|resolvable 2-design]]. 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. Strict versions include [[https://en.wikipedia.org/wiki/Kirkman%27s_schoolgirl_problem|Kirkman's Schoolgirl Problem]] (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://oeis.org/search?q=schoolgirl&sort=&language=german&go=Suche|oeis]]} === 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. === Upper Version === The "[[https://github.com/fpvandoorn/Dagstuhl-tables|Dagstuhl Happy Diner problem]]" is the version where everyone meets at least once. {[[https://oeis.org/A318240|oeis]]} [[https://www.researchgate.net/publication/227715273_Equitable_resolvable_coverings|Equitable Resolvable coverings]] also seem to be a more strict form, where we can allow people to meet at most twice. {[[https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.10024|ref]]} If we have people sitting at round tables and only interacting with their neighbours, then we have the more difficult [[https://en.wikipedia.org/wiki/Oberwolfach_problem|Oberwolfach Problem]] ---- part of [[category mathematics]] Please fill all the letters into the box to prove you're human. Please keep this field empty: SavePreviewCancel Edit summary Note: By editing this page you agree to license your content under the following license: CC Attribution-Share Alike 4.0 International table_seating.1623916138.txt.gz Last modified: 2021-06-17 07:48by timbo