hello
My name is Vineet George. I have done extensive research on combination and permutation and found consistent and uniform results in my research. This results which I have found is written down on a book known as junction (an art of counting combination and permutation). In this sight I will describe one of the method to find permutation in a convenient way.
This method is known as the advance method of finding permutation.
let have a look on the arrangement of the permutation given below.
Permutation order number 
Permuted objects 
1 
a 
b 
c 
d 
2 
b 
a 
c 
d 
3 
a 
c 
b 
d 
4 
b 
c 
a 
d 
5 
c 
a 
b 
d 
6 
c 
b 
a 
d 
7 
a 
b 
d 
c 
8 
b 
a 
d 
c 
9 
a 
c 
d 
b 
10 
b 
c 
d 
a 
11 
c 
a 
d 
b 
12 
c 
b 
d 
a 
13 
a 
d 
b 
c 
14 
b 
d 
a 
c 
15 
a 
d 
c 
b 
16 
b 
d 
c 
a 
17 
c 
d 
a 
b 
18 
c 
d 
b 
a 
19 
d 
a 
b 
c 
20 
d 
b 
a 
c 
21 
d 
a 
c 
b 
22 
d 
b 
c 
a 
23 
d 
c 
a 
b 
24 
d 
c 
b 
a 
Conclusion: From above permutation and while experiencing the formation of permutation of 4P4 we conclude that
1) For odd PON (permutation order number) the first two previously selected objects (a and b), a is previous to b. For even PON considering the first two previously selected object (a and b) b is previous to a.
2) c steps down its position by its previous position after remaining in previous position for P (max) (permutation max. i.e. NPN) of two objects with respect to the previous objects (a and b). That is c steps down its position after remaining for 2P2=2 in the same vertical position and remain for another 2P2 in the new vertical position.
3) d steps down its position by its previous position after remaining in previous position for P(max) of three objects with respect to the previous objects (a b and c). That is d steps down its position after remaining for 3P3=6 in the same vertical position and remain for another 3P3=6 in the new vertical position.
By experiencing the formation of permutation we can conclude that we can find the permutation of any number of objects without actually finding the order of permutation of previous lower number of objects as it is required in other permutations.
Now let us find the 4P4 permutation by applying the above conclusion.
Let the objects be a b c d and its selected position is a= 1^{st} position object, b=2^{nd} position object, c= 3^{rd} position object, d= 4^{th} position object.
1 





2 

d remains for P(max) of 3 objects= 3P3=6 and changes its position and remain for another P(max) of 3 objects in the new position. 

c remains for P(max) of 2 objects= 2P2=2 and changes its position and remain for another P(max) of 2 objects in the new position. 

Permutation order number 
Permuted objects 

Permutation order number 
Permuted objects 

1^{st} linear position 
2^{nd} linear position 
3^{rd} linear position 
4^{th} linear position 


1^{st} linear position 
2^{nd} linear position 
3^{rd} linear position 
4^{th} linear position 
1 



d 

1 


c 
d 
2 



d 

2 


c 
d 
3 



d 

3 

c 

d 
4 



d 

4 

c 

d 
5 



d 

5 
c 


d 
6 



d 

6 
c 


d 
7 


d 


7 


d 

8 


d 


8 


d 

9 


d 


9 


d 

10 


d 


10 


d 

11 


d 


11 


d 

12 


d 


12 


d 

13 

d 



13 

d 


14 

d 



14 

d 


15 

d 



15 

d 


16 

d 



16 

d 


17 

d 



17 

d 


18 

d 



18 

d 


19 
d 




19 
d 



20 
d 




20 
d 



21 
d 




21 
d 



22 
d 




22 
d 



23 
d 




23 
d 



24 
d 




24 
d 














3 





4 




similarly placing c in all the remaining sections we get. 

For odd (PON) a is previous to b. For even (PON) b is previous to a 

Permutation order number 
Permuted objects 

Permutation order number 
Permuted objects 

1^{st} linear position 
2^{nd} linear position 
3^{rd} linear position 
4^{th} linear position 


1^{st} linear position 
2^{nd} linear position 
3^{rd} linear position 
4^{th} linear position 
1 


c 
d 

1 
a 
b 
c 
d 
2 


c 
d 

2 
b 
a 
c 
d 
3 

c 

d 

3 
a 
c 
b 
d 
4 

c 

d 

4 
b 
c 
a 
d 
5 
c 


d 

5 
c 
a 
b 
d 
6 
c 


d 

6 
c 
b 
a 
d 
7 


d 
c 

7 
a 
b 
d 
c 
8 


d 
c 

8 
b 
a 
d 
c 
9 

c 
d 


9 
a 
c 
d 
b 
10 

c 
d 


10 
b 
c 
d 
a 
11 
c 

d 


11 
c 
a 
d 
b 
12 
c 

d 


12 
c 
b 
d 
a 
13 

d 

c 

13 
a 
d 
b 
c 
14 

d 

c 

14 
b 
d 
a 
c 
15 

d 
c 


15 
a 
d 
c 
b 
16 

d 
c 


16 
b 
d 
c 
a 
17 
c 
d 



17 
c 
d 
a 
b 
18 
c 
d 



18 
c 
d 
b 
a 
19 
d 


c 

19 
d 
a 
b 
c 
20 
d 


c 

20 
d 
b 
a 
c 
21 
d 

c 


21 
d 
a 
c 
b 
22 
d 

c 


22 
d 
b 
c 
a 
23 
d 
c 



23 
d 
c 
a 
b 
24 
d 
c 



24 
d 
c 
b 
a 











Similarly for a given permutation for N objects
1) The last object (N) of permutation remains in the same vertical position for P(max) of second last object (N1) and step down its position to the next successive vertical and remains for another P(max) of second last object (N1) in the new vertical.
2) The second last object (N1) of permutation remains in the same vertical position for P(max) of third last object (N2) and step down its position to the next successive vertical and remains for another P(max) of third last object (N2) in the new vertical.
3) Similarly the 3^{rd} last object remains for P (max) of 4^{th} last object (N3) and change its position and remains for another P(max) of 4^{th} last object (N3) in the new vertical position.
.
.
.
.
N) For odd PON the first object will be previous to second object. For even PON the second object will be previous to first object.
If you are interested in Junction, Combination, Permutation, then email me at softlaws4095@gmail.com or mail me at softlaws.junctions4095@gmail.com 