Non-Associative Property of 123-Avoiding Class of Aunu Permutation Patterns

Abstract

This paper presents the non-associative and non-commutative properties of the 123-avoiding patterns of Aunu permutation patterns. The generating function of the said patterns has been reported earlier by the author [1] [2]. The paper describes how these non-associative and non commutative properties can be established by using the Cayley table on which a binary operation is defined to act on the 123-avoiding and 132-avoiding patterns of Aunu permutations using a pairing scheme. Our results have generated larger matrices from permutations of points of the Aunu patterns of prime cardinality. It follows that the generated symbols can be used in further studies and analysis in cryptography and game theory thereby providing an interdisciplinary approach and applications of these important permutation patterns.

The Aunu Generated [ 7 4 ]-Linear Code and the Known [7 4 3 ] Hamming Code Using the (U|u+V)

“…Introduction Historically, Claude Shannon's paper titled "A Mathematical theory of Communication" in the early 1940s signified the beginning of coding theory and the first error-correcting code to arise was the presently known Hamming [7,4,3] code, discovered by Richard Hamming in the late 1940s [1]. As it is central, the main objective in coding theory is to devise methods of encoding and decoding so as to effect the total elimination or minimization of errors that may have occurred during transmission [2] due to disturbances in the channel. The special class of the (132) and (123) avoiding Patterns of AUNU permutations has found applications in various areas of applied Mathematics [3].…”

“…We consider the Carley table below, which is constructed using An(132) for n=7 [2] We now convert the entries of the Carley table above to the binary system using Modulus 2 arithmetic. Table 1…”

Aunu Integer Sequence as Non-Associative Structure and Their Graph Theoretic Properties

“…An overview of Aunu numbers, Aunu permutations patterns, the 123 and 132 avoiding patterns and their applications was reported by the authors in [2]. This paper considered the prime enumerative function ( ) , 123 , 5 n A n≥ generated by the author in [3] and defined an operator on some was closed in ( ) , 123 n A .…”

A Standard Generator/Parity Check Matrix for Codes from the Cayley Tables Due to the Non-associative (123)-Avoiding Patterns of AUNU Numbers

“…This special class of the (132) and (123)-avoiding class of permutation patterns which were first reported [4], where some group and graph theoretic properties were identified, had enjoyed a wide range of applications in various areas of applied Mathematics. In [1], we described how the non-associative and non-commutative properties of the patterns can be established using their Cayley tables where a binary operation was defined to act on the (132) and (123)-avoiding patterns of the AUNU numbers using a pairing scheme.…”