INTRODUCTION
The Special (132)avoiding Permutation Pattern was earlier on Studied
(Ibrahim and Audu, 2005; Ibrahim, 2007) in relation to some group theoretic
Properties, Graph theoretic properties and enumeration procedure as permutationavoiding
subwords. Algorithmic method was also employed (Ibrahim, 2007) in the
construction of some Eulerian circuits using the said scheme.
Algorithms are viewed, in Mathematics and computing as a set of procedure
for accomplishing some task which posses a given initial state and a well
defined end state. Algorithms have been used extensively in relation to
graphtheoretic models (Ibrahim, 2006, 2007). Further, in relation to
interconnection networks (Drager and Fettwies, 2002), algorithms have
also proved to be very useful tools.
In this study, a relationship has been established between the special
permutation pattern and cellular algebra. This procedure not only provide
a good exposition of the subject matter of pattern avoidance but also
introduces some new dimensions of theoretic interpretations of this class
of permutation patterns in terms of other mathematical structures; such
as coherent configurations.
In order for a good understanding of the paper, some basic notions are
supplied below:
Cellular Algebra
A cellular algebra otherwise called a coherent configuration, can be defined
as an algebra on square matrices of complex numbers having some finite
basis
defined on a nonempty set
and consisting of matrices whose entries are strictly 0 and 1 satisfying
the following:
• 
where, K = All 1matrix. 
• 
There exists a subset of (1) whose sum is I; the identity matrix.

• 
The set Γ_{finite} of 10 is closed under transposition.
That is, for each i, there exists i* such that (B_{i})^{T}
= B_{i*} 
• 


(3) 
Where, q_{i,j}^{k} in Eq.
3 are integers such that 1≤i, j, k≤m for some positive integer
m. These integers are called intersection numbers. 
A detailed treatment of cellular algebra can be found in notes due to
Cameron (n.d) especially in relation to association scheme and permutation
groups (http://kissme.shinshuu.ac.jp/as/).
Further, Delsarte (1973) did one of the good investigations on relationships
between association schemes and cellular algebra.
However, the approach in this research is on the use of succession schemes
to propose a study of and a relationship there from with a system of cellular
algebra.
Permutation Pattern
The listing 1,2,...,n of objects in a definite order is called a sequence.
And where a particular order of arrangement is desired, such an arrangement
becomes an ordered arrangement governed by a pattern and According to
Ibrahim (2007) each such permutation pattern σ ∈ Sn {n} naturally
results into a certain arrangement of 1,2,...,n given by:
and is referred to as the arrangement associated with a permutation pattern
of points of a nonempty set Ω.
On the other hand, permutation avoiding a particular pattern of arrangement
can be understood in the light of two sequences: a sequence π consisting
of n elements arranged in a given pattern and another sequence σ
having m elements such that m<n. In such situations σ is said
to be contained as a pattern in π provided π has a subsequence
which is order isomorphic to σ. If π does not contain σ
it is said to avoid it. The set of all the σavoiding permutations
is denoted S_{n}(σ).
ALGEBRAIC INTERPRETATIONS
The fact that this special succession scheme forms a cyclic Structure
has been reported earlier (Ibrahim, 2007). It was also reported that the
scheme could be used to construct some interesting eulerian circuits (Ibrahim,
2007).
The main point of emphasis in this communication is therefore focused
on finding out some further algebraic properties of this scheme as applied
to a five element sample and, by extension, to samples of size n where
n>5. This is achieved in this research by discovering some general
rules satisfied by the scheme for all n∈A_{n,subwords},
n = 3,4,... (Ibrahim, 2007). In particular, the research identifies some
useful transformation schemes using the established theoretic properties
of the special permutation pattern in conjunction with the algorithmic
procedure to discover some interesting interrelationships between the
different structures of cyclic elements, the special permutationavoiding
patterns and the matrices of binary coding scheme.
Finally, the study proposes an open problem in the recommendation section.
Formulation of Cyclic Structures
Consider the special (132) avoiding subwords as reported (Ibrahim, 2007).
For sake of clarity, the format of the succession is treated in the following
proposition which also establishes a grouptheoretic formalism of the
pattern as mentioned before.
Proposition One
Consider a mapping β_{n}(x) on points x of a non empty set
Δ whose cardinality is Δ = n which acts on x in
such a way as to generate a cycle of consecutive positive integers with
x as the first element of a permutation pattern whose highest integer
is the cardinality Δ = n of a nonempty set. Then, β_{n}
generates cycles which posses some interesting grouptheoretic property
and which define a special pattern avoidance when viewed as subwords.
Proof
Based on the given conditions β_{n}: Δ → (12...n)
so that β_{n}(x) = (x(x+1)...n) and inductively over all
the points of Δ.
Without loss of generality, the following cycles are generated subsequently
((x + 1) (x + 2) ...x), ((x + 2) (x + 3) ...
(x + 1)),...(n x ... (n–1)) 
The results follow since each of the identical cycles forms a two group
when paired with identity permutation (1). Further, when viewed as permutation
subwords each of the elements generated represents a different subword
which is also (132)avoiding.
Example
For sample if we consider a set Δ of size Δ = 5
we have thus:
β_{5} (1) = (1 2 3 4 5), β_{5}
(2) = (2 3 4 5 1), ..., β_{5} (5) = (5 1 2 3 4), 
Formulation of Algorithm
The following algorithm transforms the permutation patterns constructed
in the foregoing into matrix representation using a special coding scheme.
Step One
In the first row of the nxn matrix assign a 1 to the position a_{i,j}
corresponding to the value i of the ith matrix element. The rest are assigned
a 0.
Step Two
Shift the entry 1 in the subsequent rows so that the nonzero entries
shift consecutively towards subsequent positions in the right of the diagonal.
Step Three
Continue the procedure in step two until a square matrix is formed.
Step Four
End if a square matrix is formed. Otherwise go to step two.
Example
The following example uses the above algorithm to construct matrices A_{n}
for sample of size n = 5.
It follows that for an mxm matrix, the algorithm is iterated m times
so that at any stage t, the position of the nonzero entry on the first
row is shifted rightwards as a_{i,j} thereby generating m matrix
elements.
As a further illustration, consider the following proposition and the
accompanying example.
Proposition Two
Perform a transformation μ from elements of the special (132)avoiding
subword β_{n}(x) to elementsof corresponding nxn matrix representations
A_{n} defined by μ: β_{n}(x) → A_{n}.
Then, the points x,x+1,... of the nonempty set Δ correspond to the
positions A_{x},A_{x+1},... of the corresponding nxn matrix
representations A_{n} which also specify the positions a_{x,j},a_{x+1,j
} of the nonzero entries in such matrix representation.
Proof
The general form of β_{n}(x) is β_{n}(x) = (x(x
+ 1). . . n). Thus, for x = 1 we have the cycle (1,2,...n). It then follows
that the points 1,2,…n represent the positions of the nonzero entries
of the first matrix representation A_{1}.
Using similar technique we can generate the rest of the transformations.
That is, we can transform β_{n}(2) to A_{2}, β_{n}(3)
to A_{3} and subsequently.
The result follows.
APPLICATION
We state below a very interesting result linking up the enumeration
scheme of the reported permutation pattern with cellular algebra.
Theorem
Denote by An = {β_{5}(1), β_{5}(2),...,β_{n}(n)}
the set of cycles β_{n}(x) of length 5 and assume a transformation
as in proposition two. Then A_{5} forms a cellular algebra.
Proof
Use A_{1} to A_{5} as in the example provided above. Then, define
a linear space over
with A_{i}, i = 1,2,...n as the basis where A_{i} s are as given
above.
Then;
Similarly, for each i ∈ {1,...,n} there exist i such that A_{i}^{T}
= A_{i}. This follows readily since each A_{i} a is square
matrix coupled with the fact that the sum total of all the five matrices
gives rise to Eq. 3 above.
As a counter example, suppose for some i ∈ {1, ..., n}, A_{i}^{T}
produces a matrix containing a 2 as an entry so that a_{s,j }=
2 say, for some s ∈ Ω. This will imply that for some matrix
A_{s} s ∈ Ω one of the entriesis a 2 contradicting the
definition of A_{i} hence, the result.
CONCLUSION
The special (132)avoiding pattern of permutation under study has proved
to have some useful algebraictheoretic properties. In this study, more
properties of this pattern have been investigated through some transformations
using binary coding scheme and based on a designed algorithm.
RECOMMENDATION AND OPEN PROBLEM
More transformations should be investigated using this special succession
scheme so as to further reveal more of the useful properties and applications.
In addition, researches should focus on revealing more interrelationships
of this permutation pattern and other algebraic structures. For instance,
more relationship can be investigated in relation to semi groups and loops.