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**7**

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### 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 ...

**2**

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**1**answer

185 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 ...

**9**

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110 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 ...

**3**

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**0**answers

123 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 ...

**3**

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**2**answers

319 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**

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**1**answer

652 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 ...

**11**

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**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**

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**1**answer

385 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.
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