#Homework by Mike Murr
#OK to post
#hw26

read(`C26.txt`);

# First Jacobi-Trudi formula: building my understanding by programming simple cases.

with(LinearAlgebra);

JTfirst_1_1_x_2 := Determinant([[hkn(1,2,x), hkn(2,2,x)],[hkn(0,2,x),hkn(1,2,x)]]);

print(JTfirst_1_1_x_2);

evalb(JTfirst_1_1_x_2 = Sc([1,1],x,2));

JTfirst_2_1_x_2 := Determinant([[hkn(2,2,x), hkn(3,2,x)],[hkn(0,2,x),hkn(1,2,x)]]);

evalb(JTfirst_2_1_x_2 = Sc([2, 1], x, 2));

JTfirst_any_any_x_2 := proc(L);
Determinant([[hkn(L[1],2,x), hkn(L[1]+1,2,x)],[hkn(L[2]-1,2,x),hkn(L[2],2,x)]]);

end proc;

JTfirst_any_any_x_2([2,1]);

JTfirst_any_any_x_2([3,2]);


#Function hkn below, modified from C26.txt

#Full homog. symmetric polynomial
#h_k(x[1],...,x[n]):
#coeff of t^k 1/(1-x[1]*t)/(1-x[2]*t)...
#weight enumerator of all subests of 
#1<= i1<=i2 <=i3<= ...<=ik <=n
#weight({2,3,5})=x[2]*x[3]*x[5]:
#hkn(2,3,x)= weight({[1,2],[1,3],[2,3],[1,1],[2,2],[3,3]))=
#x[1]*x[2]+x[1]*x[3]+x[2]*x[3]+x[1]^2+x[2]^2+x[3]^2
#Take S any subset of k elements of {1,...,n}
#Case 1: n does not belong to S
#hkn(k,n-1,x)
#Case 2: n does belong
#x[n]*hkn(k-1,n,x):
#hkn(k,n,x)=
hkn:=proc(k,n,x) option remember:
if k=0 then
  RETURN(1):
fi:
if n=0 then
  RETURN(0):
fi:

# added next 3 lines to use hkn in Jacobi-Trudi first formula
if k<0 then
  RETURN(0);
fi:

expand(hkn(k,n-1,x)+x[n]*hkn(k-1,n,x)):
end:

#* * * * * * * * * * * * * * * * * * * * * * *

#Here is the actual first Jacobi-Trudi formula

JTfirst := proc(L,m) local n, i, j;
n:=nops(L);
Determinant([seq([seq(hkn(L[i]+ j - i,m,x), j=1..n)],i=1..n)]);

end proc;

evalb(JTfirst([2,1],2) = Sc([2,1],x,2));

evalb(JTfirst([2,1],3) = Sc([2,1],x,3));

evalb(JTfirst([3,2],3) = Sc([3,2],x,3));

evalb(JTfirst([1,1,1],2) = Sc([1,1,1],x,2));

evalb(JTfirst([5,4,3,3],5) = Sc([5,4,3,3],x,5));