#Please do not post homework
#Ravali Bommanaboina, 10/11/20, Assignment 10

#Question 1
#P1 = [3,8,4,9,1,5,6,7,2]
#P2 = [8,5,6,4,3,9,7,2,1]
# 1,2,3,4,5,6,7,8,9  1,2,3,4,5,6,7,8,9
#(3,8,4,9,1,5,6,7,2)(8,5,6,4,3,9,7,2,1)
#P1*P2 = [6,2,4,1,8,3,9,7,5]
# 1,2,3,4,5,6,7,8,9  1,2,3,4,5,6,7,8,9
#(8,5,6,4,3,9,7,2,1)(3,8,4,9,1,5,6,7,2)
#P2*P1 = [7,1,5,9,4,2,6,8,3]

#inverse
#P = [2, 4, 1, 5, 8, 9, 7, 3, 6]
#  1,2,3,4,5,6,7,8,9
# (2,4,1,5,8,9,7,3,6)
#  2,3,1,5,8,9,7,3,6
# (1,2,3,4,5,6,7,8,9)
#P^(-1) = [3, 1, 8, 2, 4, 9, 7, 5, 6]

#cycle structure
#P := [4, 6, 9, 1, 3, 7, 5, 8, 2] -> [[4, 1], [8], [9, 2, 6, 7, 5, 3]]
#  1,2,3,4,5,6,7,8,9
# (4,6,9,1,3,7,5,8,2)
#4->1
#6->7->5->3->9->2
#8

#number of inversions
#P = [5, 7, 1, 4, 9, 8, 3, 6, 2] ->20
#5 precedes 1,4,3,2 (4)
#7 precedes 1,4,3,6,2 (5)
#1 precedes nothing (0)
#4 precedes 3,2 (2)
#9 precedes 8,3,6,2 (4)
#8 precedes 3,6,2 (3)
#3 precedes 2 (1)
#6 precedes 2 (1)
#2 precedes nothing (0)

#major index -> 21
#P = [5, 7, 1, 4, 9, 8, 3, 6, 2]
# descents
#7->1 (2)
#9->8 (5)
#8->3 (6)
#6->2 (8)
#2+5+6+8

#Question 2
#enter the first permutation
InvGF:=proc(S) local i,permutations,sum:
permutations:=permute(S):
sum:=0:
for i in permutations do
sum:=inv(i)+sum:
end do:
return(sum):
end:

MajGF:=proc(S) local i,permutations,sum:
permutations:=permute(S):
sum:=0:
for i in permutations do
sum:=maj(i)+sum:
end do:
return(sum):
end:

#it is true for n=1,...,7 that InvGF(S)=MajGF(S)

#Question 3
# 9, 8, 25/3, 54432/5, 7/864

#Question 4
#Question 5