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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.

Non-associative algebraic structures arise in many situations. Cayley octonions are a notorious example, but there are far more; for example, nonassociative loop arise in cordinatization of projective planes and the Einstein velocity addition in relativity theory also forms a nonassociative loop. Self distributive algebras appear naturally in the study of Braids [

The 123-avoiding class of the Aunu permutation patterns which have been found to be of both combinatorics and group theoretic importance [

Non-associative structures include structures like groupoids, quasigroup and loops, nonassociative semi-rings as well as self distributive algebras and mediality.

The oldest and most developed discipline of nonassociative algebra originated in 1930s in works of Sushkevich, Moufang, Bol, Mordorch, and others, see [

Recent researches appear in the Proceedings of International Conferences on Nonassociative Algebra and its Applications [

In order to make this paper more self-contained, some notation overview is here under presented of some key concepts used in the paper.

An arrangement of the objects

which is called the arrangement associated with a permutation pattern

Given a sequence

It is useful to differentiate between a subsequence and a subword. For instance, if

Determination of

It was reported by [

The basic procedure for generating the special permutation patterns under study, have already been outlined, see for instance [

As a pairing scheme involving pairs of numbers associated by some precedence relation [

The elements are paired in order of precedence

where

The precedence parameter acts on the elements to produce pairs such as are related as; element and first successor, element and second successor, up to element and

Under the given condition, it is required that the

where

The enumeration scheme involves doubt regarding the identity of the first element in the desired pair. Moreover, absolute certainty is desired that by the end of the enumeration, the required pair, whichever it is, is achieved.

We now state an important theorem for the enumeration of these permutation patterns.

Theorem 2.1

The number of subwords for the permutation patterns under study is enumerated as: 2,3,5,5,8, ∙∙∙ corresponding to the length (cardinality) of the special (123)-avoiding sequences 5,7,11,13,17, ∙∙∙ [

Proof

To prove this let us suppose a permutation

We now rewrite these numbers in the form of sequence as follows:

We now define a mapping

ported in theorem 2.1; and n is prime greater than or equal to five.

Then, for

where i enumerates the cycles formed in permutations of elements of

An illustrative Example is provided thus: for n = 5, a permutation can be generated for

without loss of generality, subsequent cycles can be constructed using similar procedure by rearrangement of elements of

It follows that

The following Tables 1-5 provide summarized results for the aforementioned procedure on

We now use the entries of the Cayley table to test non-associativity of points in Aunu permutations patterns of

It can be seen from the above Cayley table that, it is closed under

i.e.

Also,

It can be shown from

1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

1 | 1 | 3 | 5 | 2 | 4 |

2 | 1 | 4 | 2 | 5 | 3 |

3 | 1 | 5 | 4 | 3 | 2 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | |
---|---|---|---|---|---|---|---|

1 | 1 | 3 | 5 | 7 | 2 | 4 | 6 |

2 | 1 | 4 | 7 | 3 | 6 | 2 | 5 |

3 | 1 | 5 | 2 | 6 | 3 | 7 | 4 |

4 | 1 | 6 | 4 | 2 | 7 | 5 | 3 |

5 | 1 | 7 | 6 | 5 | 4 | 3 | 2 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 3 | 5 | 7 | 9 | 11 | 2 | 4 | 6 | 8 | 10 |

2 | 1 | 4 | 7 | 10 | 2 | 5 | 8 | 11 | 3 | 6 | 9 |

3 | 1 | 5 | 9 | 2 | 6 | 10 | 3 | 7 | 11 | 4 | 8 |

4 | 1 | 6 | 11 | 5 | 10 | 4 | 9 | 3 | 8 | 2 | 7 |

5 | 1 | 7 | 2 | 8 | 3 | 9 | 4 | 10 | 5 | 11 | 6 |

6 | 1 | 8 | 4 | 11 | 7 | 3 | 10 | 6 | 2 | 9 | 5 |

7 | 1 | 9 | 6 | 3 | 11 | 8 | 5 | 2 | 10 | 7 | 4 |

8 | 1 | 10 | 8 | 6 | 4 | 2 | 11 | 9 | 7 | 5 | 3 |

9 | 1 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 2 | 4 | 6 | 8 | 10 | 12 |

2 | 1 | 4 | 7 | 10 | 13 | 3 | 6 | 9 | 12 | 2 | 5 | 8 | 11 |

3 | 1 | 5 | 9 | 13 | 4 | 8 | 12 | 3 | 7 | 11 | 2 | 6 | 10 |

4 | 1 | 6 | 11 | 3 | 8 | 13 | 5 | 10 | 2 | 7 | 12 | 4 | 9 |

5 | 1 | 7 | 13 | 6 | 12 | 5 | 11 | 4 | 10 | 3 | 9 | 2 | 8 |

6 | 1 | 8 | 2 | 9 | 3 | 10 | 4 | 11 | 5 | 12 | 6 | 13 | 7 |

7 | 1 | 9 | 4 | 12 | 7 | 2 | 10 | 5 | 13 | 8 | 3 | 11 | 6 |

8 | 1 | 10 | 6 | 2 | 11 | 7 | 3 | 12 | 8 | 4 | 13 | 9 | 5 |

9 | 1 | 11 | 8 | 5 | 2 | 12 | 9 | 6 | 3 | 13 | 10 | 7 | 4 |

10 | 1 | 12 | 10 | 8 | 6 | 4 | 2 | 13 | 11 | 9 | 7 | 5 | 3 |

11 | 1 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 |

2 | 1 | 4 | 7 | 10 | 13 | 16 | 2 | 5 | 8 | 11 | 14 | 17 | 3 | 6 | 9 | 12 | 15 |

3 | 1 | 5 | 9 | 13 | 17 | 4 | 8 | 12 | 16 | 3 | 7 | 11 | 15 | 2 | 6 | 10 | 14 |

4 | 1 | 6 | 11 | 16 | 4 | 9 | 14 | 2 | 7 | 12 | 17 | 5 | 10 | 15 | 3 | 8 | 13 |

5 | 1 | 7 | 13 | 2 | 8 | 14 | 3 | 9 | 15 | 4 | 10 | 16 | 5 | 11 | 17 | 6 | 12 |

6 | 1 | 8 | 15 | 5 | 12 | 2 | 9 | 16 | 6 | 13 | 3 | 10 | 17 | 7 | 14 | 4 | 11 |

7 | 1 | 9 | 17 | 8 | 16 | 7 | 15 | 6 | 14 | 5 | 13 | 4 | 12 | 3 | 11 | 2 | 10 |

8 | 1 | 10 | 2 | 11 | 3 | 12 | 4 | 13 | 5 | 14 | 6 | 15 | 7 | 16 | 8 | 17 | 9 |

9 | 1 | 11 | 4 | 14 | 7 | 17 | 10 | 3 | 13 | 6 | 16 | 9 | 2 | 12 | 5 | 15 | 8 |

10 | 1 | 12 | 6 | 17 | 11 | 5 | 16 | 10 | 4 | 15 | 9 | 3 | 14 | 8 | 2 | 13 | 7 |

11 | 1 | 13 | 8 | 3 | 15 | 10 | 5 | 17 | 12 | 7 | 2 | 14 | 9 | 4 | 16 | 11 | 6 |

12 | 1 | 14 | 10 | 6 | 2 | 15 | 11 | 7 | 3 | 16 | 12 | 8 | 4 | 17 | 13 | 9 | 5 |

13 | 1 | 15 | 12 | 9 | 6 | 3 | 17 | 14 | 11 | 8 | 5 | 2 | 16 | 13 | 10 | 7 | 4 |

14 | 1 | 16 | 14 | 12 | 10 | 8 | 6 | 4 | 2 | 17 | 15 | 13 | 11 | 9 | 7 | 5 | 3 |

15 | 1 | 17 | 16 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 |

E.g.

Also,

It can be seen from

Also,

It can also be shown from

E.g.

Also

It can be seen that the above structure is non-associative and non-commutative.

e.g

Also,

This can be stated in a general form as, taken any

It follows that the permuted structures of Aunu scheme give rise to non-associative structures where points in Aunu permutations are regarded as elements of the derived sets in relation to pairing scheme modulo n, where n is necessarily a prime

Aminu AlhajiIbrahim,Sa’idu IsahAbubakar, (2016) Non-Associative Property of 123-Avoiding Class of Aunu Permutation Patterns. Advances in Pure Mathematics,06,51-57. doi: 10.4236/apm.2016.62006