Questions tagged [latin-square]
For questions about latin squares, latin rectangles, their enumeration, their properties, generalisations and related combinatorial configurations such as MOLS (sets of Mutually Orthogonal Latin Squares).
53 questions
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What is the name of this combinatorial object and place to read about it?
The title is admittedly noninformative but I could not figure out how to squeeze into it the description of the object I am interested in. Judge by yourself.
I have an alphabet on $d$ symbols. I want ...
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Symmetric polynomial from graphs
Let $g$ be a directed, connected multigraph, on $n$ vertices, without loops.
Define
$$P_g(x_1,\dots,x_n) := Sym\left[ \prod_{(i,j) \in g} (x_i-x_j) \right]$$
where $(i,j)$ is the directed edge ...
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Does there exist a Latin square of order 9 for which its 9 broken diagonals and 9 broken antidiagonals are transversals?
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 ...
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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 ...
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Converse of Frobenius
Enumerate the elements of a finite group $G$ as follows: $g_1,g_2,\dots,g_n$. Introduce $n$ variables indexed by the elements of $G$: $x_{g_1},\dots,x_{g_n}$.
Consider the matrix $X_G$ with entries $...
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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 ...
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Smallest counterexample to Stein's conjecture?
An equi-$n$-square is an $n$ by $n$ array of cells filled with the symbols $1,2,\dots,n$ so that each symbol occurs exactly $n$ times.
(Every Latin square of order $n$ is an equi-$n$-square, but the ...
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Latin squares with one cycle type?
Cross posting from MSE, where this question received no answers.
The following Latin square
$$\begin{bmatrix}
1&2&3&4&5&6&7&8\\
2&1&4&5&6&7&8&3\\...
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Determinant of symmetric Latin square
Let $n=2m$ be an even number. Let us construct $n\times n$ symmetric matrices $S_n$ in the following way. The entries are indeterminates $X_1,\ldots,X_{n-1}$. We choose a $1$-factorization of the ...
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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 ...
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(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 ...
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Is recognizing if a Latin square is isotopic to its transpose more efficient than computing its symmetry group?
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). ...
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Do successive maximum permutations pick latin squares uniformly?
Suppose we start with a $n\times n$ matrix with entries sampled independently and uniformly at random from $[0,1]$. The weight of a set of entries will simply be the sum of those entries. A ...
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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 ...
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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 ...
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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 & \...
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Coloring in Combinatorial Design Generalizing Latin Square
I have a question about a combinatorial design very similar to a Latin Square, which is arising out of an open problem in graph theory. The design is an $n \times n$ matrix whose entries we want to ...
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Dinitz Conjecture extension to rectangles
The Dinitz Conjecture, which was proved later in a more general form by Galvin, stated that given an $n\times n$ array, its elements could be filled exactly like a latin square, where the elements in ...
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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 $k$-...
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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 $j$-...
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Bounding the number of orthogonal Latin squares from above
As is usual, let $N(n)$ denote the maximum size of a set of mutually orthogonal Latin squares of order $n$. I am wondering what results hold that bound $N(n)$ from above; the only ones I can think of ...
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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 ...
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Number of solutions and minimal clues in Sixy Sudoku
Sixy Sudoku is a variation on Latin squares and traditional sudoku played on a $6 \times 6$ grid with an initial clue of several cells filled in with a subset of the digits $1$–$6$. The task is to ...
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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 ...
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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(...
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Has the existence of a 3-MOLS(10) containing a self-orthogonal Latin square and its transpose been eliminated?
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 ...
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Does every $n\times n\times n$ Latin cube contain a Latin transversal?
In 1967 H. J. Ryser conjectured that every Latin square of odd order has a Latin transversal. Similar to Latin squares, we may consider Latin cubes.
QUESTION: Let $n$ be any positive integer. Does ...
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Evans conjecture for symmetric latin squares
The Evans conjecture ( which was proved later by Smetaniuk) states that for any $n$, if at most $n-1$ entries of a partial $n\times n$ latin square are filled, it can be completed to the full latin ...
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A graph $G$ with two $K_6$ subgraphs, in which any one-factor of $G$ induces a one-factor in exactly one of the $K_6$ subgraphs?
I'm seeking a simple graph $G$ of the following type:
It contains two disjoint copies of $K_6$ (the complete graph on 6 nodes), $H$ and $H'$ say.
Any one-factor of $G$ must contain either (a) a one ...
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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 ...
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Proper Latin sub-squares of generalized Latin squares
Say we have a generalization of a Latin square, where the square is of size $n \times n$, $n = ab$ and each row and each column has $b$ occurrences of each of $[1, \dots, a]$. Is there always ...
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Is counting Latin squares #P-complete?
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 &...
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How to get Latin squares from a finite group and a subgroup
Let G be a finite group and we know its group table is a Latin square of order |G|. Now let H be any subgroup of G of index n. Then we can form G/H which is a collection of left cosets. My question is,...
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Transformation between latin squares
Let $L\in R^{k\times k}$ a Latin square matrix.
Which is the most general form of $A\in R^{k\times k}$ such that
$$
A^TLA=L'
$$
with $L'$ another Latin square?
Thanks!
Fabio
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Are there any studies about general lexicographical orderings of Latin Squares and random walks on the space of all such orderings of a given order?
Are there any previous studies about the general lexicographical orderings of Latin squares including random walks the space of all such orderings for a given order of Latin squares?
Are there any ...
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The edge precoloring extension problem for complete graphs
Consider coloring the edges of a complete graph on even order. This can be seen as the completion of an order $n$ symmetric Latin square except the leading diagonal. My question pertains to whether we ...
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graph built from orthogonal Latin Squares
I've asked the following question on MathExchange site, with a bounty, with no answer or comments. Maybe I would have additional comments here. The problem came to be while reading some articles on ...
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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 \...
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Proof that a pandiagonal Latin square of order $n$ exists iff $n$ is not a multiple of $2$ or $3$?
A pandiagonal Latin square of order $n$ is an assignment of the numbers $\{0,\ldots,n-1\}$ to the cells of an $n \times n$ grid such that no row, column, or diagonal of any length contains the same ...
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Tighter lower bound of the lower triangular sum of an arbitrary Latin square
In this math.stackexchange.com question I seek a tighter bound than the one I presented in there in the question. Rob Pratt puts forth a conjecture in his answer motivated by the dual problem of the ...
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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 ...
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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 ...
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Comparability of elements in a Latin square based on a few rows
Let $\Pi=\{\pi_1,\pi_2,\dots,\pi_n\}$ be the rows of an $n\times n$ Latin square (the order of the rows does not matter).
Each row $\pi_i$ induces an order $\prec_i$ on the elements of $[1,n]$, where $...
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Existence of latin squares with an involutory symmetry
Let $M \in \mathbb{N}$ and let $\pi \in S_{M}$ be an involution with at least one fixed point. I'm interested in finding a latin square $A$ of order $M$ such that $A_{i,j} = \pi(A_{j,i})$ for each $i,...
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A bound on the number of partial transversals of a latin square
A Latin Square (LS) of order $n$ is an $n$ on $n$ matrix, each entry contains one of the symbols $1,2,\ldots,n$, and every row and every column contains each symbol exactly once. A (complete) ...
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Is there a way to estimate the number of Latin squares with a given autotopism?
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 ...
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Unexpected non-uniformity of results from some implementations of Jacobson-Matthews seem to show a strange sensitivity to isotopy class
Questions
Why do some Jacobson-Matthews (J-M) implementations for generating random latin squares exhibit frequencies inconsistent with an underlying uniform distribution?
Further investigation ...
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What are the limits to the lengths of the sequences of consecutive completed Sudoku when order 9 Latin squares are generated in lexicographic order?
Question: What are the maximum and minimum lengths of the sequences of consecutive completed Sudoku which occur when order 9 Latin squares are generated in (standard) lexicographic order?
A minimum ...
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The condition for mutually orthogonal Latin square
Suppose $A$ and $B$ are Latin squares of order $n$. And suppose any column of $A$ and any column of $B$ have common entry only once. Then are $A$ and $B$ mutually orthogonal?
I know the converse is ...
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Is the (left or right) Bol property Isotopy-invariant?
It is well known that a loop satisfies both the left Bol property $(x(yx))z = x(y(xz))$ and the right Bol property $((zx)y)x = z((xy)x)$ if and only if it is a Moufang loop. It is also well known that ...
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