#Kafung Mok
#Homework 3
#Feb.2, 2014

Help:=proc():
print(`inv1(p), gf(n,q)`):
end:

############
#Problem 1.#
############

#f:='f':
#y:=f(x):
#dy:=diff(y,x):

#plot(sin(x),x=-2*Pi..2*Pi):

#with(plots):
#implicitplot(x^3+4*y,x=-2..1,y=-1..2);

#plot3d(exp(-x^2+y^2-1)^3), x=-1..1,y=-2..2);

#A:=matrix(2,3,[1,2,3,4,5,6]);
#B:=matrix(3,2,[2,4,5,2,1,11]);
#A&*B;
#evalm(");


############
#Problem 2.#
############

#I tried writing a program to do this, but was 
#unsuccessful. Here are the results by using Maple.
#
#n:=50
#Digits:=74 
#  
#n:=10
#Digits:=12
#
#n:=20
#Digits:=26
#evalf(det(Hil(20))), det(evalf(Hil(20)));
#
#n:=30
#Digits:=43
#
#n:=40
#Digits:=59
#
#n:=60
#Digits:=88
#
#n:=70
#Digits:=104
#
#n:=80
#Digits:=117
#
#n:=90
#Digits:=133
#
#n:=100
#Digits:=148


##############
#Problem 3 a.#
##############

#Counts the number of inversions in list p. 
inv1:=proc(p) local i,j,n,k:
n:=nops(p):
k:=0:

for i from 1 to n-1 do
 for j from i+1 to n do

    if p[i] > p[j] then
        k:=k+1:
    fi:

 od:
od:

k:
end:

##############
#Problem 3 b.#
##############

#Sum of q^inv1(p) over all the permutations of length n.
with(combinat):
gf := proc(n, q) local a, k, q1, i:

a := permute(n): 
k := nops(a): 
q1 := 0:

for i to k do 
   q1 := q^inv1(op(i, a))+q1:
od:

q1:

end: