scholarly journals Aztec Diamonds and Baxter Permutations

10.37236/377 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Hal Canary
Keyword(s):  
Permutation Matrix ◽  
Enumerative Combinatorics ◽  
Alternating Sign Matrices ◽  
Domino Tilings ◽  
Permutation Matrices ◽  
Pattern Avoiding Permutations ◽  
The Relationship

We present a proof of a conjecture about the relationship between Baxter permutations and pairs of alternating sign matrices that are produced from domino tilings of Aztec diamonds. It is shown that a tiling corresponds to a pair of ASMs that are both permutation matrices if and only if the larger permutation matrix corresponds to a Baxter permutation. There has been a thriving literature on both pattern-avoiding permutations of various kinds [Baxter 1964, Dulucq and Guibert 1988] and tilings of regions using dominos or rhombuses as tiles [Elkies et al. 1992, Kuo 2004]. However, there have not as of yet been many links between these two areas of enumerative combinatorics. This paper gives one such link.

scholarly journals Permutation Matrices, Their Discrete Derivatives and Extremal Properties

2020 ◽  
Vol 48 (4) ◽  
pp. 719-740
Author(s):  
Richard A. Brualdi ◽  
Geir Dahl
Keyword(s):  
Permutation Matrix ◽  
Permutation Matrices ◽  
Discrete Derivative ◽  
The Relationship ◽  
Derivatives Of ◽  
Extremal Properties

AbstractFor a permutation π, and the corresponding permutation matrix, we introduce the notion of discrete derivative, obtained by taking differences of successive entries in π. We characterize the possible derivatives of permutations, and consider questions for permutations with certain properties satisfied by the derivative. For instance, we consider permutations with distinct derivatives, and the relationship to so-called Costas arrays.

scholarly journals Transposable and symmetrizable matrices

1980 ◽  
Vol 29 (4) ◽  
pp. 469-474 ◽  
Author(s):  
David McCarthy ◽  
Brendan D. McKay
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Permutation Matrix ◽  
Permutation Matrices ◽  
Symmetrizable Matrices ◽  
Necessary And Sufficient ◽  
Square Matrix

AbstractA square matrix A is transposable if P(RA) = (RA)T for some permutation matrices p and R, and symmetrizable if (SA)T = SA for some permutation matrix S. In this paper we find necessary and sufficient conditions on a permutation matrix P so that A is always symmetrizable if P(RA) = (RA)T for some permutation matrix R.

scholarly journals Linear maps leaving invariant subsets of nonnegative symmetric matrices

2003 ◽  
Vol 68 (2) ◽  
pp. 221-231 ◽  
Author(s):  
Hanley Chiang ◽  
Chi-Kwong Li
Keyword(s):  
Linear Transformation ◽  
Permutation Matrix ◽  
Symmetric Matrices ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Linear Maps ◽  
Doubly Stochastic Matrices ◽  
Low Dimensional

Let  be a certain set of nonnegative symmetric matrices, such as the set of symmetric doubly stochastic matrices or the set, of symmetric permutation matrices. It is proven that a linear transformation mapping  onto  must be of the form X ↦ PtX P for some permutation matrix P except for several low dimensional cases.

scholarly journals Enumeration of Parallelograms in Permutation Matrices for Improved Bounds on the Density of Costas Arrays

10.37236/5610 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Christopher N. Swanson ◽  
Bill Correll, Jr. ◽  
Randy W. Ho
Keyword(s):  
Permutation Matrix ◽  
Permutation Matrices

A Costas array of order $n$ is an $n\times n$ permutation matrix such that all vectors between pairs of ones are distinct. Thus, a permutation matrix fails to be a Costas array if and only if it contains ones that form a (possibly degenerate) parallelogram. In this paper, we enumerate parallelograms in an $n\times n$ permutation matrix. We use our new formulas to improve Davies's $O(n^{-1})$ result for the density of Costas arrays.

scholarly journals New Cryptosystem Based-on Permutation Matrix

2018 ◽  
Vol 4 (1) ◽  
Author(s):  
Rooya Karimnia ◽  
Ghassan Khaleel ◽  
SHERZOD TURAEV
Keyword(s):  
Random Selection ◽  
Permutation Matrix ◽  
Main Concern ◽  
Permutation Matrices ◽  
And Diffusion ◽  
Confusion And Diffusion ◽  
Selection Of

Cryptography is defined as a technique of transmitting information in a secretive manner so that only authorized people are able to read and process it. The main aim of cryptography is to make sure the message has been received by the receiver in a secure manner, and not being understood by anyone else but the receiver. Although there exist several different cryptosystems, the choice of the best algorithm is the main concern of the researches. In this paper, a new cryptography system based on matrix permutation has been introduced. A permutation matrix is an nï‚´ n matrix which is obtained by permuting its rows and columns according to some permutations of the numbers 0 to n. In this paper, encryption and decryption methodologies have been proposed based on permutation matrices. The algorithms are based on the random selection of the permutation matrix - known as key matrix - entries. The main objective of this research is to evaluate the security and the performance of the proposed cryptosystem as well as to promote the confusion and diffusion.

The Maximum Number of Disjoint Permutations Contained in a Matrix of Zeros and Ones

1964 ◽  
Vol 16 ◽  
pp. 729-735 ◽  
Author(s):  
D. R. Fulkerson
Keyword(s):  
Bipartite Graphs ◽  
Permutation Matrix ◽  
Identity Matrix ◽  
Permutation Matrices ◽  
Image Position

A well-known consequence of the König theorem on maximum matchings and minimum covers in bipartite graphs (5) or of the P. Hall theorem on systems of distinct representatives for sets (4) asserts that an n by n (0, 1)-matrix A having precisely p ones in each row and column can be written as a sum of p permutation matrices:Our main objective is a generalization of (1.1) along the following lines. Let A be an arbitrary m by n (0, 1)-matrix. Call an m by n (0, 1)-matrix P a permutation matrix if PPT = I, where PT is the transpose of P and I is the identity matrix of order m.

TRIANGULAR MONOIDS AND AN ANALOG TO THE DERIVED SEQUENCE OF A SOLVABLE GROUP

2000 ◽  
Vol 10 (03) ◽  
pp. 309-321
Author(s):  
CRAIG BAUER
Keyword(s):  
Solvable Group ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Permutation Matrices ◽  
The Relationship

A sequence is defined for a monoid consisting of upper triangular matrices. This sequence is analogous to the derived sequence of the solvable group. The relationship between the two sequences is investigated and sets of upper triangular partial permutation matrices, which arise naturally from this relation, are examined.

scholarly journals A New Symmetric Cryptosystem Based on of Permutation Matrices

2016 ◽  
Vol 2 (2) ◽  
Author(s):  
M. Shafriezal ◽  
Md Kasim ◽  
A. Sofiyyullah Razalai ◽  
Ghassan Khaleel ◽  
SHERZOD TURAEV
Keyword(s):  
Permutation Matrix ◽  
Permutation Matrices ◽  
And Performance ◽  
Selection Of

A new symmetric cryptosystem based on permutation matrices is proposed. In this paper, the encryption and decryptionalgorithms were built upon random multiple selection of the elements in the permutation matrix. This technique promotes theconfusion, diffusion, and increases the complexity of proposed algorithms. The security and performance of the proposedcryptosystem is evaluated.

scholarly journals A bijection between permutations and a subclass of TSSCPPs

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Jessica Striker
Keyword(s):  
Distributive Lattice ◽  
Alternating Sign Matrices ◽  
Plane Partitions ◽  
Permutation Matrices ◽  
Inversion Number ◽  
International Audience ◽  
Natural Statistics

International audience We define a subclass of totally symmetric self-complementary plane partitions (TSSCPPs) which we show is in direct bijection with permutation matrices. This bijection maps the inversion number of the permutation, the position of the 1 in the last column, and the position of the 1 in the last row to natural statistics on these TSSCPPs. We also discuss the possible extension of this approach to finding a bijection between alternating sign matrices and all TSSCPPs. Finally, we remark on a new poset structure on TSSCPPs arising from this perspective which is a distributive lattice when restricted to permutation TSSCPPs.

scholarly journals THE APPLICATION OF ISOMORPHIC MATRIX REPRESENTATIONS FOR MODELING THE PROTOCOL FOR THE FORMATION OF SECRET KEYS-PERMUTATIONS OF HUGE SIZES

2021 ◽  
Vol 295 (2) ◽  
pp. 78-88
Author(s):  
VLADIMIR KRASILENKO ◽  
◽  
NATALIYA YURCHUK ◽  
DIANA NIKITOVICH ◽  
◽  
...  
Keyword(s):  
Permutation Matrix ◽  
Affine Transformations ◽  
Matrix Representations ◽  
Software Module ◽  
Black And White ◽  
Permutation Matrices ◽  
The Matrix ◽  
Secret Keys ◽  
Software Modules ◽  
Fixed Matrix

A The article considers the peculiarities of the application of isomorphic matrix representations for modeling the protocol of matching secret keys-permutations of significant dimension. The situation is considered when for cryptographic transformations of blocks with a length of 256 * 256 bytes, presented in the form of a matrix of a black-and-white image, it is necessary to rearrange all bytes in accordance with the matrix keys. To generate a basic matrix key and the appearance of the components KeyA and KeyB in the format of two black and white images, a software module using engineering mathematical software Mathcad is proposed. Simulations are performed, for example, with sets of fixed matrix representations. The essence of the protocol of coordination of the main matrix of permutations by the parties is considered. Also shown are software modules in Mathcad for accelerated methods that display the procedure of iterative permutations in a permutation matrix isomorphic to the elevation of the permutation matrix to the desired degree with a certain side, corresponding to specific bits of bits or other code representations of selected random numbers. It is demonstrated that the parties receive new permutation matrices after the first step of the protocol, those sent to the other party, and the identical new permutation matrices received by the parties after the second step of the protocol, ie the secret permutation matrix. Similar qualitative cryptographic transformations have been confirmed using the proposed representations of the permutation matrix based on the results of modeling matrix affine-permutation ciphers and multi-step matrix affine-permutation ciphers for different cases when the components of affine transformations are first executed in different sequences. , and then permutation using the permutation matrix, or vice versa. The model experiments performed in the study demonstrated the adequacy of the functioning of the models proposed by the protocol and methods of generating a permutation matrix and demonstrated their advantages.

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