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table_seating [2021-06-17 07:48] timbotable_seating [2022-04-04 03:32] (current) nik
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 ==== Table Seatings ==== ==== Table Seatings ====
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 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,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).  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.+More generally we want a [[https://en.wikipedia.org/wiki/Block_design#Resolvable_2-designs|resolvable 2-design]]. 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://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]]}. This is the question as to resolvable (v,3,1) 2-designs, which if I understand it, exist iff v = 3 mod 6.
  
 +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'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 whole week) {[[https://oeis.org/search?q=schoolgirl&sort=&language=german&go=Suche|oeis]]}+Table size 4: Resolvable (v,k,1)- 2-designs.  
 +[[https://www.semanticscholar.org/paper/The-spectrum-of-resolvable-designs-with-block-size-Vasiga-Furino/364fb4a75a38493ed2c86fa3589adfee6d2714f5|This paper]] says that nessesary numerical conditions are sufficient except for 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 "[[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: 2022-04-04 03:32
  • by nik