#C5.txt
Help5:=proc(): print(`WtE(S,f,x), RF(x,n),LtoR(pi) `):
print(`inv(pi), maj(pi) `):
end:

with(combinat):

#maj(pi): The major index : The sum of the places where
#pi[i]>pi[i+1]
maj:=proc(pi) local n,i,co:
co:=0:
n:=nops(pi):
for i from 1 to n-1 do
 if pi[i]>pi[i+1] then  
   co:=co+i:
 fi:
 od:
 co:
 end:
 
 

#inv(pi): The number of inversions of the permutation pi
#For example inv([1,2,3])=0, inv([3,2,1])=3  

inv:=proc(pi) local n,i,j,co:
n:=nops(pi):
co:=0:
for i from 1 to n do 
 for j from i+1 to n do   
    if pi[i]>pi[j] then 
      co:=co+1:
    fi:
od:
od:
co:
end:
	



#LtoR(pi): The list places that are larger than anything
#to the left pi=[2,1,4,3]
LtoR:=proc(pi) local n,L,i,ma:
n:=nops(pi):
L:=[1]:
#ma is the current world record
ma:=pi[1]:
for i from 2 to n do
 if pi[i]>ma then
   L:=[op(L), i ]:
   ma:=pi[i]:
 fi:
 od:
 L:
end:

#x*(x+1)*...*(x+n-1)
RF:=proc(x,n) local i:
mul(x+i,i=0..n-1):
end:



#WtE(S,f,x): add(x^f(s), s in S)
WtE:=proc(S,f,x) local s:
add(x^(f(s)), s in S)
end:

##start old stuff from C3.txt
#C3.txt, Jan. 29, 2026
Help3:=proc(): print(`FP(pi), Der(n), d(n) , ExtractCycle(pi,i) , CycDec(pi) `): end:

#FP(pi): The number of fixed points of per. pi
FP:=proc(pi) local n, i,co:
n:=nops(pi):

co:=0:
for i from 1 to n do
 if pi[i]=i then
   co:=co+1:
 fi:
od:
co:
end:

FP([2,1,4,3,5,6]);
 

# Der(n): The set of derangements (i.e. permutations w/o fixed points) of {1, ..., n}
Der := proc(n) local S,pi,DE:
S:=permute(n):
DE:={}:
for pi in S do
 if FP(pi)=0 then 
   DE:=DE union {pi}:
 fi:
od: 
DE:
end:

#a(n):=nops(permute(n))
#
#Solutions SOl. 
#a(n)-n*a(n-1)=0, a(0)=1 HOMOG. linear (first-order) recurrence
#
#d(n):=|Der(n)|
#

d:=proc(n) option remember: if n=1 then 0 else n*d(n-1)+(-1)^n fi:end:

#ExtractCycle(pi,i): The cycle corresponding to the member i of the permutation pi
#For example ExtractCycle([2,1,4,3],1)=[1,2]
ExtractCycle:=proc(pi,i) local C,ng:
C:=[i]:
ng:=pi[i]:
while ng<>i do
C:=[op(C),ng]:
ng:=pi[ng]:
od:
C:
end:

#CycDec(pi): The full cyclic decomposition of pi
CycDec:=proc(pi) local n,i,StillToDo,S,C,ng:
n:=nops(pi):
StillToDo:={seq(i,i=1..n)}:
S:={}:
while StillToDo<>{} do
 ng:=StillToDo[1];
 C:=ExtractCycle(pi,ng):
 S:=S union {C}:
#S\T : S minus T 
StillToDo:=StillToDo minus {op(C)}:
od:
S:
end: