There are asymptotically the same number of Latin squares of each parity

A Latin square is reduced if its first row and column are in natural order. For Latin squares of a particular order n there are four possible different parities. We confirm a conjecture of Stones and Wanless by showing asymptotic equality between the numbers of reduced Latin squares of each possible parity as the order n → ∞

Two classes of minimal generic fundamental invariants for tensors

Motivated by the problems raised by B\"{u}rgisser and Ikenmeyer, we discuss two classes of minimal generic fundamental invariants for tensors of order 3. The first one is defined on $\otimes^3 \mathbb{C}^m$, where $m=n^2-1$. We study its construction by obstruction design introduced by B\"{u}rgisser and Ikenmeyer, which partially answers one problem raised by them. The second one is defined on $\mathbb{C}^{\ell m}\otimes \mathbb{C}^{mn}\otimes \mathbb{C}^{n\ell}$. We study its evaluation on the matrix multiplication tensor $\langle\ell,m,n\rangle$ and unit tensor $\langle n^2 \rangle$ when $\ell=m=n$. The evaluation on the unit tensor leads to the definition of Latin cube and 3-dimensional Alon-Tarsi problem. We generalize some results on Latin square to Latin cube, which enrich the understanding of 3-dimensional Alon-Tarsi problem. It is also natural to generalize the constructions to tensors of other orders. We illustrate the distinction between even and odd dimensional generalizations by concrete examples. Finally, some open problems in related fields are raised.Comment: Some typos were changed.New publication information has been update