Questions tagged [latin-square]

For questions on or pertaining to Latin squares.

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35 views

Is there an algorithm to generate size $n$ mutually orthogonal latin squares for $n \neq 2, 6$?

Is there an algorithm to generate size $n$ mutually orthogonal latin squares for $n \neq 2, 6$? For reference, I've been scanning several texts on combinatorics (Knuth's "The Art of Computer ...
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25 views

Is there any basis for ‪Transversal‬ matrix space?

Given That An N-By-N Matrix A, Is A Set Of Numbers With One From Each Row And Each Column Is Called A Transversal Matrix Given that an n-by-n matrix A, is a set of numbers with one from each row and ...
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1answer
39 views

Why is the formal definition of Latin square equivalent with the informal?

Informally, a latin square is a table where each element appears exactly once in each row and each column. I know that this is probrably not an official definition of, however, it should somehow match ...
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Is there a proof that the total number of idempotent elements over all quasigroups of order n equals the number of quasigroups of that order?

To clarify, as requested by Community I am looking for a proof that the total number of idempotent elements over all quasigroups of order n equals the number of quasigroups of that order. Could be ...
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Combinatorial design related to scheduling group activities (everyone tries every activity, no pair is together twice)

Trying to solve this problem led me to consider the following generalization. Let $g$ and $p$ be positive integers. Imagine that you own $g$ distinct board games, where each game requires exactly $p$ ...
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1answer
92 views

How many $5\times5$ Latin squares are symmetric?

There are 161,280 Latin squares of dimension 5. I am trying to reduce this number to the number of essentially different grids -- for enumerating the Japanese puzzle Futoshiki. To preserve the ...
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Latin Square and Graph Theory

How many "$1$" can be selected at most, provided that only one from each row and column of the attached matrix $A$ is selected?(That is, when one "$1$" is selected, no other ...
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1answer
32 views

Formula for number of latin rectangles with 2 rows

So i am preparing for exam in cominatorics and reading a textbook, and this is the task can't solve: Find formula for normalized 2 row latin rectangle number $L(n,2)$ The problem is I am really stuck ...
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1answer
65 views

Explanation for the $\frac{(n!)^{2n}}{n^{n^2}}$ lower bound on the number of Latin squares?

Wikipedia says that there are at least $\frac{(n!)^{2n}}{n^{n^2}}$ Latin squares of size $n$. But the citation is a paywalled textbook. How does one prove this bound?
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Example of a pair of non-isomorphic quasi-groups with parastrophic Latin squares?

A Latin square $\Lambda$ over an alphabet $A$ is a set of triples of elements of $A$ such that for every $\alpha,\beta\in A$, there is exactly one $\gamma\in A$ for which $(\alpha,\beta,\gamma)\in \...
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Prove there is only one main class latin square of order 4.

I'm learning about latin squares and orthogonal latin squares. My question is how can I prove there's only one main class latin square of order 4? I did this one $$\array{0&1&2&3\\1&2&...
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How to prove that two Latin squares $P$ and $Q$ are orthogonal if and only if $P^{-1}Q$ is a Latin square?

The inspiration of my question comes from Henry B. Mann's 1942 paper The Construction of Orthogonal Latin Squares. My goal is to add rigor to Mann's explanation that two Latin squares $P = (P_1,P_2,......
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orthogonal latin square on division

$n$ is an odd integer and $L$ and $L^{\prime}$ are two $n \times n$ matrix define like this. in $L$ element on row $i$ and column $j$ is $(i+j) \mod n$. in $L^{\prime}$ element on row $i$ and column $...
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1answer
46 views

An example of an algebraic loop which has different L and R inverses?

Can anyone point me toward a simple example of a non-associative algebraic loop (i.e. a quasigroup with an identity) for which at least one element has a left inverse which is not equal to its right ...
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How to find the expected sum of squares for a Latin Square

having recently discovered Latin Squares, I wondered how or if it is possible to calculate the expected sum of squares for each column, row and treatment. I have looked at the example of what the sum ...
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60 views

Quickly generating random Latin squares without aid of a computer

I'm looking for an easy/fast way of generating a random (alphabetic) 26x26 Latin square "by hand". So assume your tools are something along the lines of a 30 sided letter die, and/or a bunch ...
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1answer
131 views

Self-orthogonal Latin squares

A Latin square $A$ is called self-orthogonal if $A$ and $A^{T}$ are orthogonal Latin squares. Use the elements of $\;\mathbb{Z}_v$ as the names of the rows and columns of your Latin square. Let $\...
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1answer
32 views

For which a and b does latin square exist?

Latin square $L$ is an $n\times n$ array filled with $n$ different symbols, each occurring exactly once in each row and exactly once in each column. Les us $L = l_{ij} = (a i + b j) \mod n$ For which $...
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Guessing colored hats without repetition

This entire question is inspired by Problem $12082$ of the Problems and Solutions section of the American Mathematical Monthly (see the May $2020$ issue for the solution to said problem). First, I ...
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1answer
107 views

generalization of derangements

Derangement is defined here on wikipedia as a permutation without fixed points. Consider the following generalization: an n-derangment of an m-set is a an n by m matrix in which each cell is a number ...
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1answer
79 views

Minimum (maximum) number of swaps in Latin squares

Take any $n\times n$ Latin square, $L_n$, and count the number of swaps needed to order each row, and sum them together to get $S_n$. By number of swaps, I refer to this algorithm or similar. Question:...
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Prove that matrix A so that $a_{ij}$ = (m+1)*(i+j) mod n is a symmetric Latin Square with different elements on its diagonal (m is a freely chosen Z)

It's obvious that it's symmetric because $a_{\left(i+1\right)j}=\left(m+1\right)\left(i+1+j\right) = a_{i\left(j+1\right)}=\left(m+1\right)\left(i+j+1\right)$, but how can I prove that it's a Latin ...
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Explain why $4$ cannot be replaced by $5$ in part a)

a) Construct a Latin square of order $8$ in which the submatrix formed from the first $4$ rows and $4$ columns is the addition table for $Z_4$. b) Explain why $4$ cannot be replaced by $5$ in part a) ...
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Prove Per(A)=1 if and only if A is a permutation matrix

Assume A is a doubly stochastic matrix. I was only able to show if A is a permutation matrix, then per(A)=1. I wonder how to prove conversely? Thanks a lot.
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Grid coordinate multiplicities of orthogonal Latin squares

Let $l$ be a prime power and $r$ a positive integer with $r<l$. There are known ways to construct a set of $r$ mutually orthogonal Latin squares (MOLS) of order $l$; the size of each Latin square'...
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Matrix Multiplication for Latin Squares

This is for a project, so I could really use some insight! We are working with Latin Squares, and below is the information regarding the data: Now, I don't understand what to do with Y=Xθ+e. I can'...
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1answer
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A question asking to prove an array is not a Latin square

I am trying exercises of Ch. 10 of Combinatorics by Richard Brualdi, and I am struck on this question. Let $n$ be a positive integer and let $r$ be a nonzero integer in $Z_n$ such that $\gcd(r, n)\...
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1answer
249 views

Graph built from orthogonal Latin squares

Reminder : Given a set $S$ of $n$ elements (we will use $[n]$ in the following for simplicity), a Latin square $L$ is a function $L : [n]\times [n] \to S$, i.e., an $n\times n$ array with elements in $...
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1answer
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Modifying circulant Latin Squares

Question: Given a $N \times N$ circulant Latin square, $M$, is there a sequence of algorithmic modifications that one can make to $M$ such that the main diagonal will consist of exactly $2$ distinct ...
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1answer
288 views

Is a $4×4$ Latin square possible whose trace-sum is 7 or 9 or 13 or 15?

A Latin square with trace-sum $11$ is : $$\begin{bmatrix} 1&4&2&3\\ 2&3&4&1\\ 4&1&3&2\\ 3&2&1&4\end{bmatrix}$$ Is trace sum $7/9/5/13/15$ possible in ...
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47 views

Why are orthogonal Latin Square defined the way they are?

Two Latin square of the same size are said to be orthogonal if you form a square by superimposing the two squares in the following way, $\left[\begin{array}{l}1&2&3\\3&1&2\\2&3&...
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1answer
123 views

Show that any $n \times n$ array based on $\{1,2,..,n\}$ is a Latin Square if and only if it is simultaneously orthogonal to R and C.

I've been reading Design Theory by Zhe-Xian Wan, but I have been stuck on where to begin with this question - Page 95, Exercises 4.7, Question 4.1. Could someone please give me a hint? Let \begin{...
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120 views

Bipartite matching with degree and "budget" constraint

Consider a bipartite graph of vertices of people $P_1,P_2,\dots,P_p$ with edges connected to vertices of tasks tasks $T_1,T_2,\dots,T_t$. An edge from $P_i$ to $T_j$ means that the person $P_i$ can do ...
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1answer
39 views

Finding the largest minimum

Let $A$ be the $5\times 5$ matrix $\begin{bmatrix}11& 17 & 25 & 19 & 16\\ 24& 10 & 13 & 15 &3 \\ 12& 5 & 14 & 2 & 18\\ 23 & 4 & 1 & 8 &...
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1answer
89 views

How to form a row complete Latin squares of order odd?

Hello dear mathematicians How to form a row complete latin square of order odd. I found some answers for Latin squares of order even. Regards
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How many representations of a non-commutative groups from $n$-dimensional Latin Squares

I would like to know if there is a method to determine how many $n \times n$ Latin square define the same non-commutative group $G$. This is how many representations of $G$ can be obtained from $n$ ...
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1answer
267 views

Double Latin squares

I was messing around with a mostly-unrelated crypto problem, and I encountered this puzzle: Let's define a "double Latin square": 1) It uses two-digit numbers, that start with 1-n, and also have a ...
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A sudoku-like grid with pairs of numbers

It's easy to generate a 12x12 grid such that: Each row contains the numbers 1 - 12 Each column contains the numbers 1 - 12 No row or column contains the same number twice I am trying to determine if ...
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1answer
32 views

Equivalence between Latin squares

I have two Latin squares of order 6. Is there any way to check whether they are isomorphic? I mean any program or online tool? $ L_1= \left[ {\begin{array}{cccccc} 1 & 2 & 3 & 4 &...
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1answer
129 views

Isomorphism between Latin Squares [closed]

When we say that two Latin sqaures of order n are isomorphic or when they belong to the same main class? I am particularly working in $6 \times 6 $ Latin square.
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Complete set coset representatives of a subgroup $H$ in a group $G$

Let $G$ be a group and $H$ be a finite subgroup of index $n$ in $G$. Is there any systematic way of finding a complete set of coset representatives(That is a set of coset representatives for all ...
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Latin square from group subgroup

Let $G$ be a group and $H$ be a subgroup of $G$ of index $n$. Then from the cosets $G/H$ we can construct a Latin square of order $n$ using the complete set of representatives of $H$ in $G$. Is it ...
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1answer
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3d permutation matrices

$8 \times 8$ permutation matrices correspond to patterns of 8 rooks on a chessboard with exactly 1 rook in each row or column, never 2. Consider patterns of $n^2$ "3d rooks" in an $n \times n \times ...
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1answer
152 views

Combinatorics and Latin squares

Let's have two Latin squares, in this case, the two which are shown in this question. If we superimpose them we get 34 different combinations, with two repetitions, namely 4B and 1E. Can someone find ...
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1answer
79 views

Tighter lower bound of the lower triangular sum of an arbitrary Latin square

This is a generalization of this question.In a "restricted" Latin square consists of natural numbers $1,2, ..., n$ with each number appearing in each row, column and diagonal exactly once, what is the ...
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1answer
93 views

Minimal lower triangular sum of a $5\times 5$ Latin square

In a Latin square consists of natural numbers 1,2, ..., 5 with each number appearing in each row, column and diagonal exactly once, what is the minimal sum of the lower triangle below the main ...
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Uniqueness of non-completable latin square of size $n$

It is known that partial latin squares of order $n$ and size $n-1$ can always be completed to a latin square. I want to know if non-completable partial latin squares of order $n$, size $n$ satisfy ...
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Does $(\Bbb{R}, +)$ admit an irreducible $2$-traversal?

For a given natural number $k$, I'm going to call a subset $T$ of the plane $\Bbb{R}^2$ a $k$-traversal if, for any $x \in \Bbb{R}$, \begin{align*} k &= \operatorname{card} \{(a, b) \in T : a = x\}...
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1answer
157 views

Are these anti-circulant matrices?

Consider the matrix \begin{pmatrix}1&k+2&2&k+3&\ldots&2k+1&k+1\\k+2&2&k+3&3&\ldots&k+1&1\\\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&...
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Completion of partial latin squares

I'm reading a PDF about completing partial latin squares. The reference is https://ajc.maths.uq.edu.au/pdf/22/ocr-ajc-v22-p247.pdf There is a fact that doesn't have a proof here, and I can't find it: ...

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