On the occasion of Polymath 12 devoted to the Rota basis conjecture let me remind you about the Alon-Tarsi conjecture and test your intuition concerning a strong form of the conjecture.

The sign of a Latin square is the product of signs of rows (considered as permutations) and the signs of columns. A Latin square is even if its sign is 1, and odd if its sign is -1. It is easy to see that when $n$ is odd the numbers of even and odd Latin squares are the same.

The Alon-Tarsi Conjecture: When $n$ is even the number of even Latin square is different from the number of odd Latin square.

This conjecture was proved when $n=p+1$ and $p$ is prime by Drisko and when $n=p-1$ and $p$ is prime by Glynn. The first open case is $n=26$.

A stronger conjecture that is supported by known data is that when $n$ is even there are actually more even Latin squares than odd Latin squares. (This table is taken from the Wolfarm mathworld page on the conjecture.) ### Test your intuition: Are there more even Latin squares than odd Latin squares when $n$ is even?

A poll

The Alon-Tarsi conjecture arose in the context of coloring graphs from lists. Alon and Tarsi proved a general theorem regarding coloring graphs when every vertex has a list of colors and the conjecture comes from applying the general theorem to Dinits’ conjecture that can be regarded as a statement about list coloring of the complete bipartite graph $K_{n,n}$. In 1994 Galvin proved the Dinitz conjecture by direct combinatorial proof. See this post. Gian-Carlo Rota and Rosa Huang proved that the Alon-Tarsi conjecture implies the Rota basis conjecture (over $\mathbb R$) when $n$ is even.
Let $G$ be a graph on $n$ vertices $\{1,2,\dots, n\}$. Associate to every vertex $i$ a variable $x_i$. Consider the graph polynomial $P_G(x_1,\dots ,x_n)=\prod \{(x_i-x_j) i Alon and Tarsi considered the coefficient of the monomial $\prod_{i=1}^d x_i^{d_i}$. If this coefficient is non-zero then they showed that for every lists of colors, $d_i+1$ colors for vertex $i$, there is a legal coloring of the vertices from the lists! Alon and Tarsi went on to describe  combinatorially the coefficient as the difference between numbers of even and odd Euler orientations.