Semi-Magic Permutation: A Composition Study on the Structure

In this paper we propose a definition for a semi-magic permutation and study the composition function behavior between the magic, semi-magic and non-magic permutation using the structure defined as: ( ) Where p is a prime greater than or less than five. We equally observed that no permutations for is magic or semi-magic.


Introduction
The discovery of magic square an ancient mathematical structure lead to an interesting area of mathematics called recreational mathematics. Recreational mathematics was developed from cultural, religion, and philosophical symbols.
According to Stephens [1] the oldest magic square of order 3 by 3 appeared in an ancient Chinese literature and later in India a magic square of order 4 by 4. Durer in 1514 also constructed a magic square of order 4 by 4. Since then, the structure had attract the attention of great mathematicians and the construction of magic square of different order has been ongoing. In fact, [2] gave a generalization of magic square of order 4, [3] develop an algorithm for all magic squares of order four and Dawood, et al. [4] uses folded magic square to generalize the construction of cubes (magic cubes). Magic matrices and magic stars are resulting structures from magic square. Fanja [5] study the magic squares relative to their permutation matrix and define a magic permutation. Below are ancient magic squares, for more historical development and recent work on magic squares see Andrew [6], Nordgren [7], Ms. Rupali and Sabharwal [8], Rungratgasame, et al. [9] and Neeradha, et al. [10].
A magic square is an array of positive integers such that each raw, column and diagonals (i.e. main and cross) entries all sum to the same number called the magic constant or magic sum. Ahmed [11], report that, the magic constant for n th order general magic square starting with an integer A and with entries in an increasing arithmetic series with difference D between terms is [ ] A square matrix with a magic sum is called a -magic square. Of course, magic squares are square matrices with the properties as defined. However, if the square matrix is arranged such that the entries in each diagonal (main and cross diagonal) sum up to distinct numbers, then, the square matrix is called a semi-magic square. Magic square and semi-magic square enumeration has a long history date back at least to MacMahon, Anand et al., and Stanley [12]. According to Fanja [5], Hertzpring defined the number of magic permutations as well as the number of permutations without fixed points and without reflected point. Fanja [5], was able to propose definition for a magic permutation by considering the bijection via the use of the matrix representation of permutation. He equally show that the inverse and reflected permutation of a magic permutation are magic, and state that, there exist not a magic permutation of length for In this paper, we propose a semi-magic definition for permutations using the fixed point feature of permutations, this was possible via the matrix representation of permutations an approach employed by Fanja [5], in defining magic permutation. Also the composition behavior of the magic and semi-magic permutations of some permutations were studied.

Preliminaries
Definition: A permutation is a bijecion from a non-empty set of positive integers to itself. That is, for any set , such that is a bijection.
Example. Let . Then, ( ) is a permutation on .
Permutation derangement are permutation without fixed point while non-derangement permutations are permutation with a fixed point.
Definition: Let A, B and C be non-empty sets and and the composition of and written is the relation from A into C defined as . We note that the operation of a group is a composition function. Example: let be a permutation group of length n and permutations then the composition is defined as ( ) . Definition: Let , the permutation matrix is the matrix obtained by putting ( ), where is the standard basis vector whose component is 1. Definition: An integer is said to be a fixed point of a permutation . If We denote it with .
Example: ( ) then Definition: An integer is called a reflected point of a permutation . If We denote it as .

Example:
( ) has . Which is true by , thus is 1.
Definition: A magic permutation is a permutation whose matrix representation (i.e. permutation matrix) is a magic square of magic sum 1.
We said that an integer is a pivot point if is a fixed point as well as a reflected point. The reflected permutation of a permutation is defined by . The dihedral structure a permutation group whose elements are generated via rotation and the reflection lines of a regular n-polygon was described by Conrad [13] as a rigid motion of a regular n-gon. Consider we see in one-line notation that: and which gives the cardinality

Proposition
A permutation is said to be semi-magic if for the permutation matrix the main diagonal (upper left to lower right) sum is 1.
Proof: Let such that there exist and be the permutation matrix of then, ∑ for where Definition: A permutation is said to be semi-magic if there exist a point such that where . Explicitly, a semi-magic permutation is any permutation with a fixed point. Thus every non-deranged permutation is semi-magic.
Example: the permutation is semi-magic and is not. Remark: Every magic permutation is semi-magic but not every semi-magic is magic.

Proposition:
There exist a semi-magic permutation of length for n=3. Theorem: There exist no permutation that is magic or semi-magic. Proof: The proof is trivial from definitions above. Proposition: Let for all prime then for any magic, non-magic and semi-magic permutations of the following holds: I Composition of distinct magic equals non-magic. II Self-composition of a magic equals semi-magic. III Composition of a magic and a semi-magic equals magic. IV Self-composition of a semi-magic equals non-magic. V Self-composition of a non-magic equals non-magic. VI Composition of distinct non-magic equals non-magic.

Proof:
The proof is trivially seen from the composition table for of below.

Conclusion
The author established his findings on a permutation of prime length of the structure which is true for permutations of similar length. The dihedral structure D 4 with 8-element permutation set has (2n-3) permutations with no reflected points and has no valid magic and semi-magic composition function since, there exist no permutation with one point fixed in D 4 . This can be checked for all where n is even. With this, we recommend for study, possible existence of magic and semi-magic permutations on permutations of even length n.