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table_seating [2021-06-12 12:03]
nik
table_seating [2021-06-17 07:48] (current)
timbo
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-arranging a group of people into a number of tables so that everyone sits with everyone else.+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.
  
-A strict version is an affine plane. 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.+  * 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]]} 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]]}
  
-In less strict cases we allow people to meet more often, or not to meet.+=== 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]]} 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]]}
  • table_seating.txt
  • Last modified: 2021-06-17 07:48
  • by timbo