Help:=proc(): print(`inv(pi), des(pi), maj(pi),Cov(S,f,g)`):end:
with(combinat):

#inv(pi): inputs a permutation pi and outputs the number
#of inversions
inv:=proc(pi) local s,i,j,n:
n:=nops(pi):
s:=0:

 for i from 1 to n do
   for j from i+1 to n do
    if pi[i]>pi[j] then
         s:=s+1:
    fi:
   od:
od:
s:


end:



des:=proc(pi) local s,i:
s:=0:
for i from 1 to nops(pi)-1 do
 if pi[i]>pi[i+1] then
    s:=s+1:
 fi:
od:
s:

end:


maj:=proc(pi) local s,i:
s:=0:
for i from 1 to nops(pi)-1 do
 if pi[i]>pi[i+1] then
    s:=s+i:
 fi:
od:
s:

end:

#Av(S,f): The average of the function f on the set S
Av:=proc(S,f) local s:
add(f(s), s in S)/nops(S):
end:

#Cov(S,f,g): The covariance of f and g over S
Cov:=proc(S,f,g) local af,ag:

af:=Av(S,f): ag:=Av(S,g):

add((f(s)-af)*(g(s)-ag), s in S)/ nops(S):
end: