27,773 research outputs found
Calculation of the Number of all Pairs of Disjoint S-permutation Matrices
The concept of S-permutation matrix is considered. A general formula for
counting all disjoint pairs of S-permutation matrices as a
function of the positive integer is formulated and proven in this paper. To
do that, the graph theory techniques have been used. It has been shown that to
count the number of disjoint pairs of S-permutation matrices,
it is sufficient to obtain some numerical characteristics of all
bipartite graphs.Comment: arXiv admin note: text overlap with arXiv:1211.162
On the number of mutually disjoint pairs of S-permutation matrices
This work examines the concept of S-permutation matrices, namely permutation matrices containing a single 1 in each canonical
subsquare (block). The article suggests a formula for counting mutually
disjoint pairs of S-permutation matrices in the general case
by restricting this task to the problem of finding some numerical
characteristics of the elements of specially defined for this purpose
factor-set of the set of binary matrices. The paper describe an
algorithm that solves the main problem. To do that, every binary
matrix is represented uniquely as a n-tuple of integers.Comment: arXiv admin note: substantial text overlap with arXiv:1501.03395;
text overlap with arXiv:1604.0269
Bipartite graphs related to mutually disjoint S-permutation matrices
Some numerical characteristics of bipartite graphs in relation to the problem
of finding all disjoint pairs of S-permutation matrices in the general case are discussed in this paper. All bipartite graphs of the type
, where or are
provided. The cardinality of the sets of mutually disjoint S-permutation
matrices in both the and cases are calculated.Comment: 18 pages, 13 figures. arXiv admin note: text overlap with
arXiv:1211.162
Bipartite graphs related to mutually disjoint S-permutation matrices
Some numerical characteristics of bipartite graphs in relation to the problem
of finding all disjoint pairs of S-permutation matrices in the general case are discussed in this paper. All bipartite graphs of the type
, where or are
provided. The cardinality of the sets of mutually disjoint S-permutation
matrices in both the and cases are calculated.Comment: 18 pages, 13 figures. arXiv admin note: text overlap with
arXiv:1211.162
Bipartite graphs related to mutually disjoint S-permutation matrices
Some numerical characteristics of bipartite graphs in relation to the problem
of finding all disjoint pairs of S-permutation matrices in the general case are discussed in this paper. All bipartite graphs of the type
, where or are
provided. The cardinality of the sets of mutually disjoint S-permutation
matrices in both the and cases are calculated.Comment: 18 pages, 13 figures. arXiv admin note: text overlap with
arXiv:1211.162
On an algorithm for receiving Sudoku matrices
This work examines the problem to describe an efficient algorithm for
obtaining Sudoku matrices. For this purpose, we define the
concepts of -matrix and disjoint -matrices. The
article, using the set-theoretical approach, describes an algorithm for
obtaining -tuples of mutually disjoint matrices. We
show that in input mutually disjoint matrices, it is not
difficult to receive a Sudoku matrix
Alternating Sign Matrices and Hypermatrices, and a Generalization of Latin Square
An alternating sign matrix, or ASM, is a -matrix where the
nonzero entries in each row and column alternate in sign. We generalize this
notion to hypermatrices: an hypermatrix is an
{\em alternating sign hypermatrix}, or ASHM, if each of its planes, obtained by
fixing one of the three indices, is an ASM. Several results concerning ASHMs
are shown, such as finding the maximum number of nonzeros of an ASHM, and properties related to Latin squares. Moreover, we
investigate completion problems, in which one asks if a subhypermatrix can be
completed (extended) into an ASHM. We show several theorems of this type.Comment: 39 page
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