Magic squares, rook polynomials and permutations.
Magické čtverce a sudoku
Matematika za karetní hrou dobble
V tomto článku se zabýváme návazností populární karetní hry dobble na kombinatorické struktury. Ukazujeme, že existence dokonalých balíčků karet souvisí s existencí konečných projektivních rovin a systémů ortogonálních latinských čtverců. Dále pomocí obecnější struktury, blokových schémat, diskutujeme možnosti vytváření balíčků karet pro hry s modifikovanými pravidly. Výklad, příklady i přílohy jsou uzpůsobeny tomu, aby si čtenář mohl relativně jednoduše vytvořit vlastní karetní systémy.
Maximising the permanent of -matrices and the number of extensions of Latin rectangles.
Minimal and minimum size latin bitrades of each genus
Suppose that and are partial latin squares of order , with the property that each row and each column of contains the same set of entries as the corresponding row or column of . In addition, suppose that each cell in contains an entry if and only if the corresponding cell in contains an entry, and these entries (if they exist) are different. Then the pair forms a latin bitrade. The size of is the total number of filled cells in (equivalently ). The latin bitrade is minimal if...
Minimization of ±1 matrices under line shifts
Near-homogeneous spherical Latin bitrades
A planar Eulerian triangulation is a simple plane graph in which each face is a triangle and each vertex has even degree. Such objects are known to be equivalent to spherical Latin bitrades. (A Latin bitrade describes the difference between two Latin squares of the same order.) We give a classification in the near-regular case when each vertex is of degree or (which we call a near-homogeneous spherical Latin bitrade, or NHSLB). The classification demonstrates that any NHSLB is equal to two graphs...
Nets, -sequences, and algebraic curves over finite fields with many rational points.
New results about set colorings of graphs.
New upper bounds for the size of permutation codes via linear programming.
Nonassociative triples in involutory loops and in loops of small order
A loop of order possesses at least associative triples. However, no loop of order that achieves this bound seems to be known. If the loop is involutory, then it possesses at least associative triples. Involutory loops with associative triples can be obtained by prolongation of certain maximally nonassociative quasigroups whenever is a prime greater than or equal to or , an odd prime. For orders the minimum number of associative triples is reported for both general and involutory...
On a class of commutative groupoids determined by their associativity triples
Let be a commutative groupoid such that ; ; or . Then is determined uniquely up to isomorphism and if it is finite, then for an integer .
On a distance of groups and Latin squares
On algebras of generalized Latin squares
The main result of this paper is the introduction of a notion of a generalized -Latin square, which includes as a special case the standard Latin square, as well as the magic square, and also the double stochastic matrix. Further, the algebra of all generalized Latin squares over a commutative ring with identity is investigated. Moreover, some remarkable examples are added.
On an algorithm for a double-resolvability test.
On approximation of amenable groups by finite quasigroups.
On Balanced Room Squares and Complete Balanced Howell Rotations
On biembeddings of Latin squares.
On certain regular graphs of girth 5.
On Difference Matrices, Resolvable Transversal Designs and Generalized Hadamard Matrices.