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 .

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

The Alon-Tarsi conjecture states that for even , (Drisko 1998).

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 , 2, ... 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 , . This was proven for all of the form for prime by Drisko (1998).

Alon, N. and Tarsi, M. "Coloring and Orientations of Graphs." Combinatorica12, 125-143, 1992.Drisko,
A. A. "On the Number of Even and Odd Latin Squares of Order ." Adv. Math.128, 20-35, 1997.Drisko,
A. A. "Proof of the Alon-Tarsi Conjecture for ." Electronic J. Combinatorics5, No. 1,
R28, 1-5, 1998. http://www.combinatorics.org/Volume_5/Abstracts/v5i1r28.html.Huang,
R. and Rota, G.-C. "On the Relations of Various Conjectures on Latin Squares
and Straightening Coefficients." Disc. Math.128, 237-245, 1994.Janssen,
J. C. M. "On Even and Odd Latin Squares." J. Combin. Theory
Ser. A69, 173-181, 1995.Onn, S. "A Colorful Determinantal
Identity, a Conjecture of Rota, and Latin Squares." Amer. Math. Monthly104,
156-159, 1997.Sloane, N. J. A. Sequences A065711,
A114628, A114629,
A114630, A114631,
and A114632 in "The On-Line Encyclopedia
of Integer Sequences."Zappa, P. "The Cayley Determinant of
the Determinant Tensor and the Alon-Tarsi Conjecture." Adv. Appl. Math.19,
31-44, 1997.