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Uniform One-Factorisations
nshapeExists?Example/Reference
2[2]Yes group table
3[3]Yes group table
4[4]Noodd perm
[2,2]Yes group table
5[5]Yes group table
[3,2]Noodd perm
6[6]Noodd perm
[4,2]Yes Royle
[3,3]NoArchdeacon
[2,2,2]Noodd perm
7[7]Yes group table
[5,2]Noodd perm
[4,3]Noodd perm
[3,2,2]Yes Archdeacon
8[8]Noodd perm
[6,2]Yes Royle
[5,3]No
[4,4]No
[4,2,2]Noodd perm
[3,3,2]Noodd perm
[2,2,2,2]Yes group table
9[9]YesRoyle (64 types)
[7,2]Noodd perm
[6,3]Noodd perm
[5,4]Noodd perm
[5,2,2]Yes Royle (unique)
[4,3,2]NoRoyle
[3,3,3]Yes Royle (2 types)
[3,2,2,2]Noodd perm
10[10]Noodd perm
[8,2]?
[7,3]NoRoyle
[6,4]?
[6,2,2]Noodd perm
[5,5]NoRoyle
[5,3,2]Noodd perm
[4,4,2]Noodd perm
[4,3,3]Noodd perm
[4,2,2,2]NoRoyle
[3,3,2,2]NoRoyle
[2,2,2,2,2]Noodd perm
11[11]Yesgroup table
[9,2]No odd perm
[8,3 ]No odd perm
[7,4 ]No odd perm
[7,2,2 ]
[6,5 ]No odd perm
[6,3,2, ]
[5,4,2 ]
[5,3,3]Yes Royle
[5,2,2,2 ]No odd perm
[4,4,3 ]
[4,3,2,2 ]No odd perm
[3,3,3,2 ]No odd perm
[3,2,2,2,2 ]
12 [12]No odd perm
[10,2 ]
[9,3 ]
[8,4 ]
[8,2,2]No odd perm
[7,5 ]
[7,3,2]No odd perm
[6,6 ]
[6,4,2]No odd perm
[6,3,3]No odd perm
[6,2,2,2]
[5,5,2]No odd perm
[5,4,3]No odd perm
[5,3,2,2]
[4,4,4]No odd perm
[4,4,2,2 ]
[4,3,3,2 ]
[4,2,2,2,2]No odd perm
[3,3,3,3 ]
[3,3,2,2,2]No odd perm
[2,2,2,2,2,2]
13 [13]Yes group table
[11,2 ]No odd perm
[10,3 ]No odd perm
[9,4 ]No odd perm
[9,2,2]
[8,5 ]No odd perm
[8,3,2]
[7,6 ]No odd perm
[7,4,2]
[7,3,3]
[7,2,2,2]No odd perm
[6,5,2]
[6,4,3]
[6,3,2,2]No odd perm
[5,5,3]
[5,4,4]
[5,4,2,2]No odd perm
[5,3,3,2]
[5,2,2,2,2]
[4,4,3,2]No odd perm
[4,3,3,3]No odd perm
[4,3,2,2,2]
[3,3,3,2,2]
[3,2,2,2,2,2]No odd perm
14 [14]No odd perm
[12,2]
[11,3]
[10,4]
[10,2,2]No odd perm
[9,5]
[9,3,2]No odd perm
[8,6]
[8,4,2]No odd perm
[8,3,3]No odd perm
[8,2,2,2]
[7,7]
[7,5,2]No odd perm
[7,4,3]No odd perm
[7,3,2,2]
[6,6,2]No odd perm
[6,5,3]No odd perm
[6,4,4]No odd perm
[6,4,2,2]
[6,3,3,2]
[6,2,2,2,2]No odd perm
[5,5,4]No odd perm
[5,5,2,2]
[5,4,3,2]
[5,3,3,3]
[5,3,2,2,2]No odd perm
[4,4,4,2]
[4,4,3,3]
[4,4,2,2,2]No odd perm
[4,3,3,2,2]No odd perm
[4,2,2,2,2,2]
[3,3,3,3,2]No odd perm
[3,3,2,2,2,2]
[2,2,2,2,2,2,2]No odd perm
15 [15]
[13,2]No odd perm
[12,3]No odd perm
[11,4]No odd perm
[11,2,2]
[10,5]No odd perm
[10,3,2]
[9,6]No odd perm
[9,4,2]
[9,3,3]
[9,2,2,2]No odd perm
[8,7]No odd perm
[8,5,2]
[8,4,3]
[8,3,2,2]No odd perm
[7,6,2]
[7,5,3]
[7,4,4]
[7,4,2,2]No odd perm
[7,3,3,2]No odd perm
[7,2,2,2,2]
[6,6,3]
[6,5,4]
[6,5,2,2]No odd perm
[6,4,3,2]No odd perm
[6,3,3,3]No odd perm
[6,3,2,2,2]
[5,5,5]
[5,5,3,2]No odd perm
[5,4,4,2]No odd perm
[5,4,3,3]No odd perm
[5,4,2,2,2]
[5,3,3,2,2]
[5,2,2,2,2,2]No odd perm
[4,4,4,3]No odd perm
[4,4,3,2,2]
[4,3,3,3,2]
[4,3,2,2,2,2]No odd perm
[3,3,3,3,3]
[3,3,3,2,2,2]No odd perm
[3,2,2,2,2,2,2]Yes Archdeacon