# Questions tagged [latin-square]

For questions on or pertaining to Latin squares.

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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