Before proceeding further we make
firm with the terms
a
e d b
(Object of P-Junction)
1
4 3 2
(actual position of object in the junction)
1
2 3 4
(linear position of object in the junction)
1
5 4 2
(object number)
Formula permuting (number into
object)
Let x be the permutation order number
1)
For last object
Divide x by (r-1)!
Case1€
When x/(r-1)! = decimal number
Then select the number which is
before the decimal.
Multiply (r-1)! To the selected
number which was before the decimal.
The product obtained by
multiplication is called acquired number of particular position for last object
i.e. (a.n.p.p) r
Subtract (a.n.p.p)r with x
Then we get x- (a.n.p.p)r
=y
x is known as the number to be
reduced (a.n.p.p)r is known as the obtained reducer. y is known as
the new number to be reduced.
Case2€
When x/(r-1) = rational number
Then take the number as (rational
number-1)
Multiply (r-1)! to the number
(rational number-1)
The product obtained by
multiplication is called acquired number of particular position for last
object.
I.e. (a.n.p.p)r
Subtract (a.n.p.p)r with x
then we get
X-(a.n.p.p)r =y
X is known as the number to be
reduced, (a.n.p.p)r is known as the obtained reducer. y is known as
the new number to be reduced.
2)
For second last object.
Same procedure is applied to the new
number to be reduced.
y/(r-2)! = Either (Case1 or Case2)
Where y-(a.n.p.p)(r-1)= z
.
.
.
.
(r-1) For 2nd object.
Let the new number to
be reduced be j
Dividing j by 1!
Which will be j itself
and it will be either 2 or 1 which is a rational number.
When j=2 take the
multiplier (j-1)=(2-1)=1 according to case 2
When j=1 take the
multiplier (j-1)=(1-1)=0 according to case 2
Multiply 1! By the
occurred number to (j-1). The product which occur is the acquired number of
particular position for second object which will be 1or 0.
J-(a.n.p.p)(2)=k
Where k=1
(r) For 1st
object.
The new number to be
reduced is k
Divide k by 0!
k/0!=1 which is a
rational number.
So the multiplier will
be (1-1) =0 according to case 2
Multiply 0! By 0 The
product obtained will be acquired number of particular position for 1st
object which will be 0
k-(a.n.p.p)(1) =
l = 1
We have found all the
(a.n.p.p) for all the objects from 1 to r.
Arranging all the
(a.n.p.p) corresponding to particular position as shown we get
1)
(a.n.p.p)r for last
object. (actual position =r)
2)
(a.n.p.p)(r-1) for 2nd
last object. (actual position = r-1)
3)
(a.n.p.p)(r-2) for 3rd
last object. (actual position = r-2)
.
.
.
r-1) (a.n.p.p) for 2nd
object (actual position =2)
r) (a.n.p.p) for 1st
object. (actual position =1)
Linear Position compared to previous
object will be.
Linear position of objects in the
junction compared to previous position objet
We have found the (a.n.p.p) of each
and every object now it is the time to place each and every object according to
its linear position in the junction.
(Linear position of current
particular object with respect to previous position object)=(actual position of
object) - (acquired number of particular position)/(actual position of
object-1)!
Find all the linear position of
current particular object with respect to previous position object.
After finding the linear position as
stated above for all the objects from last to 1st. Now is the time
to place the object.
Before placing the object some
arrangement should be done.
Arrange the blank equal to the number
of objects (r) as shown below.
__ __
__ __ __
__ . . . . . . . . . . . __ __
1 2 3
4 5
6
(r-1) (r) (linear position)
Fill in the blanks on the basis of
the linear position of current particular object with respect to previous
position objects. Start placing the object from last object to 1st
object
Place the last position object on the
basis of its linear position of current particular object with respect to
previous position object.
After placing the last position
object it is the time to place the second last position object.
While placing the second last
position object only unfilled blank is taken into consideration.
Similarly place the 2nd
last object 3rd last object till the 1st object on the
basis of linear position of current particular object with respect to previous
position object on the unfilled blank space.
This will be the linear position of
objects.