**1**

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**1**answer

41 views

### How to generate a Latin square $M'$ in the same main class as $M \in \mathrm{LS}(9)$ which agrees with $L$ in the most cells?

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 \...

**5**

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**0**answers

93 views

### For which divisors $a$ and $b$ of $n$ does there exist a Latin square of order $n$ that can be partitioned into $a \times b$ subrectangles?

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 & \...

**1**

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**0**answers

48 views

### What is the minimum number of filled cells in a partial Latin rectangle with autotopism group $\cong C_2$ and autoparatopism group $\cong S_3$?

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 ...

**1**

vote

**1**answer

194 views

### Creating a Latin rectangle from a projective plane

Given a projective plane I'd like to form a latin rectangle from the lines. In particular, I'd like to take each line from the plane, order the elements in some way, and stick them into the matrix as ...

**3**

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**0**answers

63 views

### Signatures of latin squares: what about the extremal cases?

For a latin square (LS) of order $n$, we will define a cut (or maybe general transversal, I don't know whether there is an entrenched name for this) as a collection of $n$ cells such that no two share ...

**5**

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**0**answers

88 views

### Lower bound on the number of k-plexes in a Latin square

Let $A$ be an order-$n$ Latin square. A $k$-plex of $A$ is a set of entries , $k$ from each row and column and $k$ from each symbol.
My question is: Is there a Latin square with a large number of $k$-...

**3**

votes

**1**answer

131 views

### Connections between loops (algebraic structure) and graphs

I would like to know whether there are known constructions which provide a bijection between loops (isomorphism classes) and (possibly directed) graphs. Any reference to a useful paper in this ...

**7**

votes

**1**answer

215 views

### Is there a simple proof that there is no five mutually orthogonal Latin squares of order 6?

It is well known that there is a projective plane of order $n$ if and only if
there exist a set of $n-1$ mutually orthogonal Latin squares. The first nontrivial
case is $n=6$, which fails because of ...

**7**

votes

**2**answers

393 views

### How many finite loops?

How many finite loops of order $n$ are there?
I am interested in the exact values of $n$ if $n <40$ or even reasonable estimates. I am also interested in formulae or bounds for all $n$.
Note ...

**3**

votes

**1**answer

221 views

### Are all symmetric idempotent Latin squares known?

Are all symmetric idempotent Latin squares known?
There is such a square of order $n$ if and only if $n$ is odd. However, is there a classification of all of them?
(The motivation for the question ...

**11**

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**0**answers

151 views

### Proving that the set of $\lfloor n/3 \rfloor+1$ partial Latin squares given by Pebody is unavoidable?

Introduction
Cutler and Öhman (2006) attribute to Pebody (via personal communication) a construction of a set of $k:=\lfloor n/3 \rfloor+1$ partial Latin squares which are unavoidable (i.e., any ...

**4**

votes

**0**answers

146 views

### Existence problem for a generalisation of Latin squares (matrices with fixed row and column sets)

Let $R_1,\ldots,R_n$ and $C_1,\ldots,C_n$ be sets of size n.
When does there exist an $n \times n$ matrix in which the $i$-th row is a permutation of $R_i$, for all $1 \leq i \leq n$, and the $j$-...

**3**

votes

**2**answers

343 views

### What is the number of k-regular subgraphs of $K_{12,12}$?

I'm thinking about making an attempt at counting the number of Latin squares of order 12. Currently I'm in (what I call) the "sanity check" stage, where we check that the ranges that need searching ...

**7**

votes

**1**answer

667 views

### (0,1)-matrix congruence: is it known?

[[UPDATE: This work has now been published at SIAM J Discrete Math.: Formulae for the Alon–Tarsi Conjecture.]]
By equating two formulae (one congruence by Glynn (1) (which has just appeared) and one ...

**12**

votes

**0**answers

1k views

### Finding a chromatic polynomial by polynomial fitting

I would like to find the chromatic polynomial χ for the n by m rook's graph Gn,m for as many values of n and m possible. The rooks graph is also (a) the line graph of the complete bipartite graph ...

**3**

votes

**1**answer

457 views

### Diagonally-cyclic Steiner Latin squares

A Steiner triple system is a decomposition of $K_n$ into $K_3$, such as $S=\{013,026,045,124,156,235,346\}$. Steiner triple systems give rise to a Steiner Latin squares, such as $L$ below.
\[L=\left(...