Data on transversals in Latin squares

• A paper I wrote with Brendan McKay and Jeanette McLeod about the number of transversals in Latin squares.
• A paper I wrote with Judy Egan on Latin squares with resticted transversals.
• A paper I wrote with Afsane Ghafari on Latin squares with no pair of disjoint transversals and no entry common to all transversals.

If you want to know what format these files are in, it is my usual latin squares format.

• Latin squares without transversals
  • Order 2
  • Order 4
  • Order 6
  • Order 8

• Confirmed bachelor latin squares (i.e. ones with at least one entry not in a transversal). Each file is in increasing order of number of transversal-free entries.
  • Order 2
  • There are no examples of order 3
  • Order 4
  • Order 5
  • Order 6
  • Order 7
  • Order 8
  • Order 9

• Latin squares with the fewest number of transversals. For even orders we know of latin squares without transversals (see above), so this is really only of interest for odd orders.
  • Order 3 (3 transversals)
  • Order 5 (3 transversals)
  • Order 7 (3 transversals)
  • Order 9 (68 transversals)
  • Order 11 (778 transversals -- may not be the least possible, but has the fewest among the many squares I've checked)

• Latin squares with the fewest number of disjoint transversals. For even orders we know of latin squares without transversals (see above), so this is really only of interest for odd orders.
  • Order 5 (all transversals intersect)
  • Order 7 (all transversals intersect)
  • Order 9 (at most 3 disjoint transversals)

• Latin squares with the largest number of transversals.
  • Order 3 (3 transversals)
  • Order 4 (8 transversals)
  • Order 5 (15 transversals)
  • Order 6 (32 transversals)
  • Order 7 (133 transversals)
  • Order 8 (384 transversals)
  • Order 9 (2241 transversals)
  • Order 10 (5504 transversals -- provisional winner)
  • Order 11 (37851 transversals -- provisional winner)
  • Order 12 (198144 transversals -- provisional winner)
  • Order 13 (1030367 transversals -- provisional winner)
  • Order 14 (3477504 transversals -- provisional winner)
  • Order 15 (36362925 transversals -- provisional winner)
  • Order 16 (244744192 transversals -- provisional winner)
  • Order 17 (1606008513 transversals -- provisional winner)
  • Order 18 (6434611200 transversals -- provisional winner*)
  • Order 19 (87656896891 transversals -- provisional winner)
  • Order 20 (697292390400 transversals -- provisional winner)
  * N.B. The two (non-equivalent) turn-squares of order 18 given here both have 6434611200 transversals, which is the most known. The description given in our paper is incorrect, although the number of transversals was quoted correctly. These squares have turn number four. They are created by turning 4 entries in the same row of Z₉×Z₂. In the first square the 4 turned entries are consecutive. In the second there is a gap of one place before the 4th turned entry.

• Latin squares in which every set of disjoint transversals extends to a 1-partition (files are ordered by decreasing number of transversals, and orthogonal mates).
  • Order 3
  • Order 4
  • Order 5
  • There are no examples of order 6 or 7
  • Order 8

• Latin squares that contain a transversal that intersects every other transversal. Files are ordered in increasing order of the size of the largest set of disjoint transversals.
  • Order 5
  • There are no examples of order 6
  • Order 7
  • Order 8
  • Order 9

• Explicit transversals for my paper with Afsane Ghafari.
  • Transversals in G_n for 36≤n≤10000 with n≡2 mod 4. (104Mb gzipped file)
  • Transversals in H_n for 36≤n≤10000 with n≡0 mod 4. (78Mb gzipped file)
  Each line of each file lists the value of n, then a list of n columns c₀...c_n-1 with the interpretation that the transversal includes the entry in cell (i,c_i) for each i.