darij_jacobi_trudi.sagews

e = SymmetricFunctions(QQ).elementary()
h = SymmetricFunctions(QQ).homogeneous()
s = SymmetricFunctions(QQ).schur()
m = SymmetricFunctions(QQ).monomial()

for lam in Partitions(7):
#lam=Partition([2,1]);
    l=lam.length();
    mu=Partition([0]);
    entries=[];
    for i in range(l):
        for j in range(l):
            k=lam.get_part(i)-mu.get_part(j)-1;
            if k<0:k=0;
            entries.append(go_down(lam.get_part(i)-mu.get_part(j)-i+j,k));
    A=matrix(e,l, entries);
    print(lam);
    #print(A);
    print(s(A.determinant()));#," ----------- ",(A.determinant()));
    print("________________________________")

[7] s[1] + 6*s[1, 1] + 15*s[1, 1, 1] + 20*s[1, 1, 1, 1] + 15*s[1, 1, 1, 1, 1] + 6*s[1, 1, 1, 1, 1, 1] + s[1, 1, 1, 1, 1, 1, 1] ________________________________ [6, 1] s[2] + 5*s[2, 1] + 10*s[2, 1, 1] + 10*s[2, 1, 1, 1] + 5*s[2, 1, 1, 1, 1] + s[2, 1, 1, 1, 1, 1] ________________________________ [5, 2] s[2] + 4*s[2, 1] + 6*s[2, 1, 1] + 4*s[2, 1, 1, 1] + s[2, 1, 1, 1, 1] + 4*s[2, 2] + 6*s[2, 2, 1] + 4*s[2, 2, 1, 1] + s[2, 2, 1, 1, 1] ________________________________ [5, 1, 1] s[3] + 4*s[3, 1] + 6*s[3, 1, 1] + 4*s[3, 1, 1, 1] + s[3, 1, 1, 1, 1] ________________________________ [4, 3] s[2] + 3*s[2, 1] + 3*s[2, 1, 1] + s[2, 1, 1, 1] + 5*s[2, 2] + 6*s[2, 2, 1] + 2*s[2, 2, 1, 1] + 3*s[2, 2, 2] + s[2, 2, 2, 1] ________________________________ [4, 2, 1] s[3] + 3*s[3, 1] + 3*s[3, 1, 1] + s[3, 1, 1, 1] + 3*s[3, 2] + 3*s[3, 2, 1] + s[3, 2, 1, 1] ________________________________ [4, 1, 1, 1] s[4] + 3*s[4, 1] + 3*s[4, 1, 1] + s[4, 1, 1, 1] ________________________________ [3, 3, 1] s[3] + 2*s[3, 1] + s[3, 1, 1] + 3*s[3, 2] + 2*s[3, 2, 1] + s[3, 2, 2] ________________________________ [3, 2, 2] s[3] + 2*s[3, 1] + s[3, 1, 1] + 2*s[3, 2] + s[3, 2, 1] + 2*s[3, 3] + s[3, 3, 1] ________________________________ [3, 2, 1, 1] s[4] + 2*s[4, 1] + s[4, 1, 1] + 2*s[4, 2] + s[4, 2, 1] ________________________________ [3, 1, 1, 1, 1] s[5] + 2*s[5, 1] + s[5, 1, 1] ________________________________ [2, 2, 2, 1] s[4] + s[4, 1] + s[4, 2] + s[4, 3] ________________________________ [2, 2, 1, 1, 1] s[5] + s[5, 1] + s[5, 2] ________________________________ [2, 1, 1, 1, 1, 1] s[6] + s[6, 1] ________________________________ [1, 1, 1, 1, 1, 1, 1] s[7] ________________________________

def go_down(m,k):#returns e_m(1,1,...,1,x_{k+1},x_{k+2},...)
    if (m==0): return 1;
    if (m<0): return 0;
    pol=0;
    for t in range(k+1):
        pol=pol+binomial(k,t)*e([m-t]);
    return pol

︠20c85b73-1b4e-451d-95e2-ed446ea551b2︠