#C5.txt: Maple Code for Lecture 5
Help5:=proc(): print(`Redu(L), Expa(pi,S), PermuA(n),PermuI(n) `): end:

#Redu(L): inputs a list L of numbers and outputs
#The multi-set permuation of {1, ...,nops({op(L)}) where n is the number of
#number of 
#distinct members of L and the smallest number in {op(L)}
#replaced by 1, the second smallest by 2, etc.
#For example
#Redu([evalf(Pi),evalf(Pi),evalf(exp(1)),evalf(exp(1))]
#should return
#[2,2,1,1]
##Redu([evalf(Pi),evalf(exp(1)),evalf(exp(1)),5]
#=[2,1,1,3]
#
Redu:=proc(L) local S,T,i:
S:=sort(convert({op(L)},list)):

for i from 1 to nops(S) do
 T[S[i]]:=i:
od:

[seq(T[L[i]],i=1..nops(L))]:

end:

#Expa(pi,S): inputs a permutation pi of 1,..., nops(pi) and
#a sorted list of distinct numbers S, replaces 1 by
#the smallest member of S, etc. 
#For example
#Expa([2,1,3],[11,13,15])= [13,11,15]
Expa:=proc(pi,S) local i:
subs({seq(i=S[i],i=1..nops(S))},pi):
end:

#PermuA(n): inputs a non-neg. integer n and outputs the list of
#all n! permutations of {1,...,n} in lex-order (same as permute(n) in combinat)
PermuA:=proc(n) local S,L,j, J,pi:
option remember:
if n=0 then
 RETURN([[]]):
fi:

S:=PermuA(n-1):

L:=[]:


for j from 1 to n do
  #J:=1...j-1,j+1..n
  J:=[seq(1..j-1), seq(j+1..n)]:
   L:= [op(L),seq([ j, op(Expa(S[i],J)) ],i=1..nops(S))] :
od:
L:
end:

#PermuI(n): the list of all permutations of {1, ...,n} using inertion
PermuI:=proc(n)  local S,i,pi,pi1,L,j,i1:

option remember:
if n=0 then
 RETURN([[]]):
fi:


S:=PermuI(n-1):

L:=[]:

for i from 1 to nops(S) do
pi:=S[i]:

for j from 1 to n do
 pi1:=[op(1..j-1,pi),n,op(j..nops(pi),pi)]:
 L:=[op(L),pi1]:
od:
od:

L:
end: