Alon-Tarsi Conjecture

The Alon-Tarsi conjecture states that for every even order, the numbers of even and odd Latin squares differ (Drisko 1998).

A Latin square is said to be odd if it contains an odd number of rows and columns that are odd permutations. Otherwise, it is said to be even.

Let the number of even Latin squares of order be denoted , and the number of odd Latin squares of order be denoted . The following table summarizes the numbers of even and odd Latin squares for small .

Sloane	A114628	A114629	A114630
1	1	0	1
2	2	0	2
3	6	6	0
4	576	0	576
5	80640	80640	0
6	505958400	306892800	199065600
7	30739709952000	30739709952000	0
8	55019078005712486400	53756954453370470400	1262123552342016000

If is odd, then switching two rows of a Latin square alters its sign, so .

Zappa (1997) generalized the conjecture to fixed diagonal Latin squares to encompass odd orders. Define a fixed diagonal Latin square as a Latin square for which all diagonal entries equal 1, and denote the numbers of fixed diagonal even and fixed diagonal odd Latin squares of order by and , respectively. For , 2, ..., equals 1, 1, 0, 24, 384, ... (OEIS A114631), and equals 0, 0, 2, 0, 960, ... (OEIS A114632).

Further define the Alon-Tarsi constant by

(1)

(Drisko 1998). Then the values of for , 3, ... are 1, , 4, , 2304, 368640, 6210846720, ... (OEIS A065711; Drisko 1998).

The quantity is related to the numbers of even and odd Latin squares by

(2)

(Drisko 1998).

The extended Alon-Tarsi conjecture states that for every positive integer , . Drisko (1998) proved the prime-order case. His claimed extension to relied on an invalid result of Zappa and was therefore not proved by that argument (Stones and Wanless 2012).