read `C:\\Users\\jlc450\\Desktop\\C18.txt`
Help()
       Toe(f,x,n,e),  ToE(f,x,n,E), Andrew(F,x,n), mToe(n) 
Help15()
         P(n), ApplyPi(f,x,n,pi), Sym(f,x,n), mL(x,L,n)
  IsSym(f,x,n), ToM(f,x,n,M), ei(x,i,n), eL(x,L,n), eTom(n)  

# Problem 1

hni:=proc(n,i,x) local L:
 add(mL(x,L,n),L in P(i)):
end:
T1:=hni(4,3,x)
       3       2            2            2                 2
   x[1]  + x[1]  x[2] + x[1]  x[3] + x[1]  x[4] + x[1] x[2] 

                                                   2
      + x[1] x[2] x[3] + x[1] x[2] x[4] + x[1] x[3] 

                                  2       3       2     
      + x[1] x[3] x[4] + x[1] x[4]  + x[2]  + x[2]  x[3]

            2                 2                             2
      + x[2]  x[4] + x[2] x[3]  + x[2] x[3] x[4] + x[2] x[4] 

            3       2                 2       3
      + x[3]  + x[3]  x[4] + x[3] x[4]  + x[4] 

# Problem 2

Hni:=proc(n,i,x) local f,t,j:
 f:=mul(1-x[j]*t,j=1..n):
 f:=1/f:
 f:=taylor(f,t=0,i+1):
 expand(coeff(f,t,i)):
end:

T2:=Hni(4,3,x)
       3       2            2            2                 2
   x[1]  + x[1]  x[2] + x[1]  x[3] + x[1]  x[4] + x[1] x[2] 

                                                   2
      + x[1] x[2] x[3] + x[1] x[2] x[4] + x[1] x[3] 

                                  2       3       2     
      + x[1] x[3] x[4] + x[1] x[4]  + x[2]  + x[2]  x[3]

            2                 2                             2
      + x[2]  x[4] + x[2] x[3]  + x[2] x[3] x[4] + x[2] x[4] 

            3       2                 2       3
      + x[3]  + x[3]  x[4] + x[3] x[4]  + x[4] 

T2-T1
                               0

# Problem 3

hL:=proc(x,L,n):
 mul(hni(n,L[i],x),i=1..nops(L)):
end:

hL(x,[2,1],4)
/    2                                           2            
\x[1]  + x[1] x[2] + x[1] x[3] + x[1] x[4] + x[2]  + x[2] x[3]

                     2                   2\                    
   + x[2] x[4] + x[3]  + x[3] x[4] + x[4] / (x[1] + x[2] + x[3]

   + x[4])

hTom:=proc(n) local L,mat,i,f,col,l:
 L:=P(n):
 mat:=[]:
 for i from 1 to nops(L) do
  f:=hL(x,L[i],n):
  f:=expand(f):
  f:=ToM(f,x,n,M):
  col:=[seq(coeff(f,M[l]),l in L)]:
  mat:=[op(mat),col]:
 od:
 mat:
end:

mToh:=proc(n):
 inverse(hTom(n)):
end:

# Problem 4

pow:=proc(n,i,x) local j:
 add(x[j]^i,j=1..n):
end:

pL:=proc(x,L,n) local i:
 mul(pow(n,L[i],x),i=1..nops(L)):
end:

pTom:=proc(n) local L,mat,i,f,col,l:
 L:=P(n):
 mat:=[]:
 for i from 1 to nops(L) do
  f:=pL(x,L[i],n):
  f:=expand(f):
  f:=ToM(f,x,n,M):
  col:=[seq(coeff(f,M[l]),l in L)]:
  mat:=[op(mat),col]:
 od:
 mat:
end:

mTop:=proc(n):
 inverse(pTom(n)):
end:

# Problem 5

TransMatrix:=proc(n,base1,base2) local milk,cookies:
 if base1=m then
  cookies:=IdentityMatrix(nops(P(n))):
 elif base1=e then
  cookies:=eTom(n):
 elif base1=h then
  cookies:=hTom(n):
 elif base1=p then
  cookies:=pTom(n):
 fi:
 if base2=m then
  milk:=IdentityMatrix(nops(P(n))):
 elif base2=e then
  milk:=mToe(n):
 elif base2=h then
  milk:=mToh(n):
 elif base2=p then
  milk:=mTop(n):
 fi:
 multiply(milk,cookies):
end:

yoodle:=proc(n) local S,r,t:
 S:={m,e,h,p}:
 for r in S do
  for t in S minus {r} do
   print(`The transformation from `,r,` to `,s,` is:`)
   print(TransMatrix(n,r,t)):
  od:
 od:
end:

seq(yoodle(n),n=1..6)

# Problem 6

T:=proc(L) local A,j,l:
 A:={op(L[1])}:
 j:=L[2]:
 l:=L[3]:
 if member(j,A) then
  A:=A minus {j}:
  RETURN([[op(A)],j,l+1]):
 else
  A:=A union {j}:
  RETURN([[op(A)],j,l-1]):
 fi:
end:

squiggly:=proc(n,k) local S,l,sub,S1:
 S:={}:
 for l from 1 to k do
  sub:=choose([seq(i,i=1..n)],k-l):
  S1:={seq(seq([A,j,l],A in sub),j in [seq(i,i=1..n)])}:
  S:=S union S1:
 od:
 sub:=choose([seq(i,i=1..n)],k):
 S1:={seq(seq([A,j,0],j in A),A in sub)}:
 S union S1:
end:

noodle:=squiggly(3,2)
{[[], 1, 2], [[], 2, 2], [[], 3, 2], [[1], 1, 1], [[1], 2, 1], 

  [[1], 3, 1], [[2], 1, 1], [[2], 2, 1], [[2], 3, 1], 

  [[3], 1, 1], [[3], 2, 1], [[3], 3, 1], [[1, 2], 1, 0], 

  [[1, 2], 2, 0], [[1, 3], 1, 0], [[1, 3], 3, 0], [[2, 3], 2, 0], 

  [[2, 3], 3, 0]}

{seq(T(L),L in noodle)}
{[[], 1, 2], [[], 2, 2], [[], 3, 2], [[1], 1, 1], [[1], 2, 1], 

  [[1], 3, 1], [[2], 1, 1], [[2], 2, 1], [[2], 3, 1], 

  [[3], 1, 1], [[3], 2, 1], [[3], 3, 1], [[1, 2], 1, 0], 

  [[1, 2], 2, 0], [[1, 3], 1, 0], [[1, 3], 3, 0], [[2, 3], 2, 0], 

  [[2, 3], 3, 0]}

evalb(noodle=%)
                              true

spaghetti:=proc(N) local n,k:
 for n from 1 to N do
  for k from 1 to n do
   if evalb(squiggly(n,k)={seq(T(L),L in squiggly(n,k))})=false then
    RETURN(false):
   fi:
  od:
 od:
 true:
end:

spaghetti(12)
                              true