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# Name:Treasa Bency Biju Jose
# Date: 10-06-2020
# Assignment #10
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1. 

By MAPLE:
P1 := randperm(9);
               P1 := [1, 9, 6, 2, 7, 3, 8, 4, 5]

P2 := randperm(9);
               P2 := [2, 1, 4, 9, 8, 6, 7, 3, 5]

MulPers(P1, P2);
                  [2, 5, 6, 1, 7, 4, 3, 9, 8]

MulPers(P2, P1);
                  [9, 1, 2, 5, 4, 3, 8, 6, 7]

(ii) 

By MAPLE:
P := randperm(9);
                P := [4, 5, 8, 2, 1, 3, 6, 9, 7]

InvPer(P);
                  [5, 4, 6, 1, 2, 7, 9, 3, 8]

(iii) By hand:
[1, 2, 3, 4, 5, 6, 7, 8, 9]
[5, 6, 1, 9, 4, 8, 2, 3, 7]

PtoC(P) by hand: [1, 5, 4, 9, 7, 2, 6, 8, 3]

By MAPLE:
P := randperm(9);
                P := [5, 6, 1, 9, 4, 8, 2, 3, 7]

PtoC(P);
                 [[9, 7, 2, 6, 8, 3, 1, 5, 4]]

(iv) By hand:
inv(P) by hand: 15

maj(P) by hand: 9

By MAPLE:
P := randperm(9);
                P := [9, 4, 5, 1, 3, 2, 6, 7, 8]

inv(P);
                               15

maj(P);
                               9
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2. 
with(combinat);
InvGF := proc(n, q) local i, B, A;
A := 0; 
B := permute(n); 
for i to nops(B) do 
  A := A + q^inv(B[i]); 
end do; 
factor(A); 
end proc;


with(combinat);
MajGF := proc(n, q) local i, B, A; 
A := 0; 
B := permute(n); 
for i to nops(B) do 
  A := A + q^maj(B[i]); 
end do; 
factor(A); 
end proc;

Y := proc(k) local n; 
for n to 7 do 
  evalb(InvGF(n, k) = MajGF(n, k)); 
end do; 
end proc;
Y(2);
                              true
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3.
InvGF(2, 5)/InvGF(1, 5);
                               6

InvGF(3, 5)/InvGF(2, 5);
                               31

InvGF(4, 5)/InvGF(3, 5);
                              156

InvGF(5, 5)/InvGF(4, 5);
                              781

InvGF(6, 5)/InvGF(5, 5);
                              3906

InvGF(7, 5)/InvGF(6, 5);
                             19531

with(gfun);
factor(guessgf([6, 31, 156, 781, 3906, 19531], x));
                  [-(5*x - 6)/((5*x - 1)*(x - 1)), ogf]
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