Problems in Latin squares

In mathematics, the theory of Latin squares is an active research area with many open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings. Problems posed here appeared in, for instance, the Loops (Prague) conferences and the Milehigh (Denver) conferences.

Open problems

Bounds on maximal number of transversals in a Latin square

A transversal in a Latin square of order n is a set S of n cells such that every row and every column contains exactly one cell of S, and such that the symbols in S form {1, ..., n}. Let T(n) be the maximum number of transversals in a Latin square of order n. Estimate T(n).

Proposed: by Ian Wanless at Loops '03, Prague 2003
Comments: Wanless, McKay and McLeod have bounds of the form cⁿ < T(n) < dⁿ n!, where c > 1 and d is about 0.6. A conjecture by Rivin, Vardi and Zimmermann (Rivin et al., 1994) says that you can place at least exp(c n log n) queens in non-attacking positions on a toroidal chessboard (for some constant c). If true this would imply that T(n) > exp(c n log n). A related question is to estimate the number of transversals in the Cayley tables of cyclic groups of odd order. In other words, how many orthomorphisms do these groups have?

The minimum number of transversals of a Latin square is also an open problem. H. J. Ryser conjectured (Oberwolfach, 1967) that every Latin square of odd order has one. Closely related is the conjecture, attributed to Richard Brualdi, that every Latin square of order n has a partial transversal of order at least n − 1.

Characterization of Latin subsquares in multiplication tables of Moufang loops

Describe how all Latin subsquares in multiplication tables of Moufang loops arise.

Proposed: by Aleš Drápal at Loops '03, Prague 2003
Comments: It is well known that every Latin subsquare in a multiplication table of a group G is of the form aH x Hb, where H is a subgroup of G and a, b are elements of G.

Densest partial Latin squares with Blackburn property

A partial Latin square has Blackburn property if whenever the cells (i, j) and (k, l) are occupied by the same symbol, the opposite corners (i, l) and (k, j) are empty. What is the highest achievable density of filled cells in a partial Latin square with the Blackburn property? In particular, is there some constant c > 0 such that we can always fill at least c n² cells?

Proposed: by Ian Wanless at Loops '03, Prague 2003
Comments: In a paper to appear, Wanless has shown that if c exists then c < 0.463. He also constructed a family of partial Latin squares with the Blackburn property and asymptotic density of at least exp(-d(log n)^1/2) for constant d > 0.

Largest power of 2 dividing the number of Latin squares

Let $L_{n}$ be the number of Latin squares of order n. What is the largest integer $p(n)$ such that $2^{p(n)}$ divides $L_{n}$ ? Does $p(n)$ grow quadratically in n?

Proposed: by Ian Wanless at Loops '03, Prague 2003
Comments: Of course, $L_{n}=n!(n-1)!R_{n}$ where $R_{n}$ is the number of reduced Latin squares of order n. This immediately gives a linear number of factors of 2. However, here are the prime factorizations of $R_{n}$ for n = 2, ...,11:

2	3	4	5	6	7	8	9	10	11
1	1	2²	2³7	2⁶37²	2¹⁰35*1103	2¹⁷31361291	2²¹3²5231*3824477	2²⁸3²53137*547135293937	2³⁵3⁴528012206499*62368028479

This table suggests that the power of 2 is growing superlinearly. The best current result is that

R_{n}

is always divisible by f!, where f is about n/2. See (McKay and Wanless, 2003). Two authors noticed the suspiciously high power of 2 (without being able to shed much light on it): (Alter, 1975), (Mullen, 1978).

References

Alter, Ronald (1975), "How many latin squares are there?", Amer. Math. Monthly, 82 (6), Mathematical Association of America: 632–634, doi:10.2307/2319697, JSTOR 2319697.
McKay, Brendan; Wanless, Ian (2005), "On the number of latin squares", Ann. Comb., 9 (3): 335–344, doi:10.1007/s00026-005-0261-7, S2CID 7289396.
Mullen, Garry (1978), "How many i-j reduced latin squares are there?", Amer. Math. Monthly, 85 (9), Mathematical Association of America: 751–752, doi:10.2307/2321684, JSTOR 2321684.
Rivin, Igor; Vardi, Ilan; Zimmerman, Paul (1994), "The n-queens problem", Amer. Math. Monthly, 101 (7), Mathematical Association of America: 629–639, doi:10.2307/2974691, JSTOR 2974691.

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