#C18.txt, Nov. 7, 2016, Dr. Z.'s Experimental Mathematics Class, Bases of Symmetric Polynomials

Help:=proc(): print(`  Toe(f,x,n,e),  ToE(f,x,n,E), Andrew(F,x,n), mToe(n) `): end:

with(linalg):


#Stuff from C15.txt ##################

   #C15.txt: ExpMath, Rutgers, Oct. 27, 2016, Bases of the alg. of Symmetric Polynomials
   Help15:=proc(): print(`P(n), ApplyPi(f,x,n,pi), Sym(f,x,n), mL(x,L,n)`):
   print(`IsSym(f,x,n), ToM(f,x,n,M), ei(x,i,n), eL(x,L,n), eTom(n)  `):
   end:
    with(combinat):
   #Rev(L): reverse of L
   Rev:=proc(L) local n,i: n:=nops(L): [seq(L[n-i+1],i=1..n)]: end:
 
    #P(n): the list of (integer) partitons of n in non-increasing order
    P:=proc(n) local S,i:
    S:=partition(n):
    Rev([seq(Rev(S[i]),i=1..nops(S))]):
    end:

    #ApplyPi(f,x,n,pi): inputs f, x, n, pi, where f is an expression in x[1],...,x[n]
    #pos. integer n, letter x, and a permutation pi of {1, ...,n}, applies x[i]->x[pi[i]]
    #to f
    ApplyPi:=proc(f,x,n,pi) local i:
    if nops(pi)<>n then
      RETURN(FAIL):
    fi:

     subs({seq(x[i]=x[pi[i]],i=1..n)},f):
     end:
   
      #Sym(f,x,n): the symmetrizer of f
      Sym:=proc(f,x,n) local pi,S:
      S:=permute(n):
      
       convert({seq(ApplyPi(f,x,n,pi), pi in S)},`+`):


      end:

        #mL(x,L,n): The monomial symmetric function m_L(x1,..., xn) where n>=nops(L)
        mL:=proc(x,L,n) local i:
        if n<nops(L) then
         RETURN(FAIL):
        fi:

         Sym(mul(x[i]^L[i],i=1..nops(L)), x,n):
        
        end:

         #IsSym(f,x,n): inputs f and x (a lettter), f is an expression in x[1],..., x[n], n pos.int.
         #outputs true (or false) iff f is symmetric
         IsSym:=proc(f,x,n) local i:
          for i from 1 to n-1 do
           if subs({x[i]=x[i+1],x[i+1]=x[i]},f)-f<>0 then
             RETURN(false):
           fi:
          od:
           true:
          end:
           
             KickZ:=proc(L) local i:
              for i from 1 to nops(L) while L[i]<>0 do  od:
               [op(1..i-1,L)]:
              end:

                            
             
           #ToM(f,x,n,M): inputs a symmetric polynomial f in x[1]..., x[n]
            #outputs it as a linear combination of M[L]
           ToM:=proc(f,x,n,M) local F,f1,mono,L,c,i:
             
             if not IsSym(f,x,n) then
                RETURN(FAIL):
             fi:

 

              if not type(f,`+`) then
               c:=normal(f/mul(x[i],i=1..n)):
               if not type(c,integer) then
                    RETURN(FAIL):
                else
                RETURN(c*M[[1$n]]):
              fi:
             fi:
             
            f1:=f:
             F:=0:
             while f1<>0 do
               mono:=op(1,f1):
               L:=[seq(degree(mono,x[i]),i=1..n)]:
                c:=normal(mono/mul(x[i]^L[i],i=1..nops(L))):

                L:=KickZ(sort(L,`>`)):

                   F:=F+c*M[L]:
                  f1:=expand(f1-c*mL(x,L,n)):
           
              
               
              od:

             F:

            end:
       #ei(x,i,n): the elementary symmetric function e_i(x[1], ..., x[n])
      ei:=proc(x,i,n) :
       if i>n then
        0:
       else
      mL(x,[1$i],n):
      fi:

         end:
       
       #eL(x,L,n): the mul(e_L[i], i=1..n)
       eL:=proc(x,L,n) local i:
       mul(ei(x,L[i] ,n),i=1..nops(L)):
        end:

    




#eTom(n): code by JinYoung Park
eTom:=proc(n) local Pn,k,A,B,M,i,M1,j,x:

Pn:=P(n):

k:=nops(Pn):

A:=[seq(M[Pn[i]],i=1..k)]:

B:=[seq(ToM(expand(eL(x,Pn[i],n)),x,n,M),i=1..k)]:

M:=[]:

for i from 1 to nops(A) do
 M1:=[seq(coeff(B[i],A[j]),j=1..nops(A))]:
 M:=[op(M),M1]:
od:

end:



######Code from previou classes

#mToe(n): the transition matrix from the m-basis to the e-basis
mToe:=proc(n)
inverse(eTom(n)):
end:


#Andrew(F,x,n): inputs a symmetric polynomial F, in x[1],..., x[n]  abd ouputs the vector of coefficients
Andrew:=proc(F,x,n) local A,Pars,M,i:

A:=ToM(expand(F),x,n,M):
Pars:=P(n):

 [seq(coeff(A,M[Pars[i]],1),i=1..nops(Pars))]:

end:



#ToE(f,x,n,E): inputs a symmetric function f, homog. of total degree n, of x[1]. ..., x[n]
#and a positive integer n, and a symbol E, expresses it as a linear combination of
#the E-basis, where E[L] means e_L[1]*...e_L[nops(L)]
ToE:=proc(f,x,n,E) local v,Pars,i:

v:=multiply(Andrew(f,x,n),mToe(n)):
Pars:=P(n):

add(v[i]*E[Pars[i]],i=1..nops(Pars)):


end:

#Toe(f,x,n,e): inputs a symmetric function f, homog. of total degree n, of x[1]. ..., x[n]
#and a positive integer n, and a symbol e, expresses it as polynomial expression in e[1],e[2],...,e[n]
#thereby confirming the fundamental theorem of Symmetric Functions that any summetric polynomial
#in x[1],..,x[n] of degree<=n can be thus expressed

Toe:=proc(f,x,n,e) local v,Pars,i,j:

v:=multiply(Andrew(f,x,n),mToe(n)):
Pars:=P(n):

add(v[i]*mul(e[Pars[i][j]],j=1..nops(Pars[i])),i=1..nops(Pars)):

end: