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table_seating [20210617 07:48] timbo 
table_seating [20211208 08:44] (current) timbo 
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_2designsresolvable 2design]]. 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_2designsresolvable 2design]]. 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_problemKirkman'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=Sucheoeis]]}. This is the question as to resolvable (v,3,1) 2designs, which if I understand it, exist iff v = 3 mod 6. 
 
 For pairs, resolvable (v,2,1) 2designs exist only for even v, v >= 4. 
 
 Table size 4: Resolvable (v,k,1) 2designs. 
 [[https://www.semanticscholar.org/paper/ThespectrumofresolvabledesignswithblocksizeVasigaFurino/364fb4a75a38493ed2c86fa3589adfee6d2714f5This paper]] says that nessesary numerical conditions are sufficient except for a case that need not concern us. 
 
 
Strict versions include [[https://en.wikipedia.org/wiki/Kirkman%27s_schoolgirl_problemKirkman'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=Sucheoeis]]}  
 
=== Lower Version===  === Lower Version=== 