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A Latin square of order $n$ has $n$ broken diagonals and $n$ broken antidiagonals. When $n \equiv \pm 1 \pmod 6$, we have diagonally cyclic Latin squares in which those $2n$ diagonals are transversal …

My coauthors and I created a canonical labelling method for Latin squares based on partial Latin squares in our paper: Fang, Stones, Marbach, Wang, Liu, Towards a Latin-Square Search Engine (pdf), ISP …

I feel like I should know the answer to this. I did some Googling and didn't easily find the answer... Question: Is counting Latin squares #P-complete? Obviously the corresponding decision problem "I …

Oh, I just realized how to prove non-existence for all $n \equiv 3 \pmod 6$. We take the circulant and back-circulant Latin squares, defined as $L_{ij} = i+j \pmod n$ and $M_{ij} = i-j \pmod n$. Supp …

I eventually co-authored a paper which includes this topic. The non-trivial results are: There exists a Latin square of order $n$ which decomposes into $2 \times (n/2)$ subrectangles for all even $ …

There exists a Latin square of order $8$ which can be partitioned into $2 \times 4$ subrectangles: $$ \begin{bmatrix} \color{red} 1 & \color{red} 2 & \color{red} 3 & \color{red} 4 & \color{purple} 5 & …

If I understand correctly, these are symmetric frequency squares. Quoting from a random reference on the topic: An $F(n;\lambda_1,\ldots,\lambda_m)$ frequency square is an $n \times n$ array cons …

McKay, Meynert, Myrvold (2006) (Small latin squares, quasigroups, and loops, DOI:10.1002/jcd.20105, author copy) computationally eliminate the possibility of set of 3 mutually orthogonal Latin squares …

I'm brainstorming an idea for storing a compressed list of main class representatives of Latin squares of order $9$. One way to compress the list would be to store one Latin square $L_1$, and for $i …

An autotopism of a Latin square $L$ of order $n$ is a triple of permutations $(\alpha,\beta,\gamma)$ for which $L$ is stabilized after permuting the rows by $\alpha$, the columns by $\beta$, and the s …

Ihrig and Ihrig (2007) described a mathematical method for determining if a Latin square is isotopic to its transpose (where isotopic Latin squares vary by permuting the rows, columns and symbols). T …

Definitions: a partial Latin rectangle is an $r \times s$ matrix containing symbols from $[n] \cup \{\cdot\}$ such that each row and each column contains at most one copy of any symbol in $[n]$. The …

Alan Sokal proved that chromatic roots are dense in the whole complex plane. I.e., if $P(G;z)$ denotes the chromatic polynomial of a finite simple graph $G$ evaluated at $z \in \mathbb{C}$, then $$\b …