SymmetricFunctions(QQ['q','t'].fraction_field()).inject_shorthands(verbose=False) (q,t)=p.base_ring().gens() # Hilbert series for n>=2 ply2 = 1+q+t ply3 = 1+\ 2*q + 2*t+\ 2*q^2 + 3*q*t + t^2+\ q^3 + q^2*t ply4 = 1\ +3*q + 3* t\ +5 *q^2 + 8 * q* t + 3* t^2\ +6 * q^3 + 11* q^2* t + 6 * q* t^2 + t^3\ +5 * q^4 + 9 * q^3* t + 4 * q^2 *t^2\ +3 * q^5 + 4 * q^4* t + q^3 *t^2\ +q^6 + q^5* t ply5 = 1+\ 4*q + 4*t+\ 9*q^2 + 15*q*t + 6*t^2+\ 15*q^3 + 31*q^2*t + 20*q*t^2 + 4*t^3+\ 20*q^4 + 46*q^3*t + 35*q^2*t^2 + 10*q*t^3 + t^4+\ 22*q^5 + 51*q^4*t + 39*q^3*t^2 + 10*q^2*t^3+\ 20*q^6 + 44*q^5*t + 29*q^4*t^2 + 5*q^3*t^3+\ 15*q^7 + 29*q^6*t + 15*q^5*t^2 + q^4*t^3+\ 9*q^8 + 14*q^7*t + 5*q^6*t^2+\ 4*q^9 + 5*q^8*t + q^7*t^2+\ q^10 + q^9*t ply6=1+5*q + 5*t\ +14*q^2 + 24*q*t + 10*t^2\ +29*q^3 + 64*q^2*t + 45*q*t^2 + 10*t^3\ +49*q^4 + 125*q^3*t + 111*q^2*t^2 + 40*q*t^3 + 5*t^4\ +71*q^5 + 196*q^4*t + 196*q^3*t^2 + 85*q^2*t^3 + 15*q*t^4 + t^5\ +90*q^6 + 257*q^5*t + 266*q^4*t^2 + 119*q^3*t^3 + 20*q^2*t^4\ +101*q^7 + 287*q^6*t + 290*q^5*t^2 + 119*q^4*t^3 + 15*q^3*t^4\ +101*q^8 + 276*q^7*t + 258*q^6*t^2 + 89*q^5*t^3 + 6*q^4*t^4\ +90*q^9 + 229*q^8*t + 188*q^7*t^2 + 50*q^6*t^3 + q^5*t^4\ +71*q^10 + 163*q^9*t + 113*q^8*t^2 + 21*q^7*t^3\ +49*q^11 + 98*q^10*t + 55*q^9*t^2 + 6*q^8*t^3\ +29*q^12 + 49*q^11*t + 21*q^10*t^2 + q^9*t^3\ +14*q^13 + 20*q^12*t + 6*q^11*t^2\ +5*q^14 + 6*q^13*t + q^12*t^2\ +q^15 + q^14*t # Frobenius series for n>=2 sf2 = (q+t)*s[1,1]+s[2] sf3 = (q^3+q^2*t+q*t+t^2)*s[1, 1, 1] + (q^2+q*t+q+t)*s[2, 1] + s[3] sf4 = (q^6+q^5*t+q^4*t+q^3*t^2+q^3*t+q^2*t^2+q*t^2+t^3)*s[1, 1, 1, 1]\ + (q^5+q^4*t+q^4+2*q^3*t+q^2*t^2+q^3+2*q^2*t+q*t^2+q*t+t^2)*s[2, 1, 1]\ + (q^4+q^3*t+q^2*t+q*t^2+q^2+q*t)*s[2, 2] + (q^3+q^2*t+q^2+q*t+q+t)*s[3, 1] + s[4] sf5 = (q^10+q^9*t+q^8*t+q^7*t^2+q^7*t+q^6*t^2+q^6*t+2*q^5*t^2+q^4*t^3+\ q^4*t^2+q^3*t^3+q^3*t^2+q^2*t^3+q*t^3+t^4)*s[1, 1, 1, 1, 1]\ + (q^9+q^8*t+q^8+2*q^7*t+q^6*t^2+q^7+3*q^6*t+2*q^5*t^2+q^6+3*q^5*t+3*q^4*t^2+q^3*t^3\ +2*q^4*t+3*q^3*t^2+q^2*t^3+q^3*t+2*q^2*t^2+q*t^3+q*t^2+t^3)*s[2, 1, 1, 1]\ + (q^8+q^7*t+q^7+2*q^6*t+q^5*t^2+q^6+3*q^5*t+2*q^4*t^2+q^5+3*q^4*t+3*q^3*t^2+q^2*t^3\ +q^4+2*q^3*t+2*q^2*t^2+q*t^3+q^2*t+q*t^2)*s[2, 2, 1]\ + (q^7+q^6*t+q^6+2*q^5*t+q^4*t^2+2*q^5+3*q^4*t+q^3*t^2+q^4+3*q^3*t+2*q^2*t^2+q^3+2*q^2*t\ +q*t^2+q*t+t^2)*s[3, 1, 1]\ + (q^6+q^5*t+q^5+2*q^4*t+q^3*t^2+q^4+2*q^3*t+q^2*t^2+q^3+2*q^2*t+q*t^2+q^2+q*t)*s[3, 2]\ + (q^4+q^3*t+q^3+q^2*t+q^2+q*t+q+t)*s[4, 1] + s[5] sf6 = (q^15+q^14*t+q^13*t+q^12*t^2+q^12*t+q^11*t^2+q^11*t+2*q^10*t^2+q^9*t^3+q^10*t\ +2*q^9*t^2+q^8*t^3+2*q^8*t^2+2*q^7*t^3+q^7*t^2+2*q^6*t^3+q^5*t^4+q^6*t^2+2*q^5*t^3\ +q^4*t^4+q^4*t^3+q^3*t^4+q^3*t^3+q^2*t^4+q*t^4+t^5)*s[1, 1, 1, 1, 1, 1]\ + (q^14+q^13*t+q^13+2*q^12*t+q^11*t^2+q^12+3*q^11*t+2*q^10*t^2+q^11+4*q^10*t+4*q^9*t^2\ +q^8*t^3+q^10+4*q^9*t+5*q^8*t^2+2*q^7*t^3+3*q^8*t+6*q^7*t^2+3*q^6*t^3+2*q^7*t+5*q^6*t^2\ +4*q^5*t^3+q^4*t^4+q^6*t+4*q^5*t^2+4*q^4*t^3+q^3*t^4+2*q^4*t^2+3*q^3*t^3+q^2*t^4+q^3*t^2\ +2*q^2*t^3+q*t^4+q*t^3+t^4)*s[2, 1, 1, 1, 1]\ + (q^13+q^12*t+q^12+2*q^11*t+q^10*t^2+2*q^11+4*q^10*t+2*q^9*t^2+q^10+5*q^9*t+5*q^8*t^2\ +q^7*t^3+2*q^9+6*q^8*t+6*q^7*t^2+2*q^6*t^3+q^8+5*q^7*t+8*q^6*t^2+4*q^5*t^3+q^7+4*q^6*t\ +6*q^5*t^2+4*q^4*t^3+q^3*t^4+2*q^5*t+5*q^4*t^2+4*q^3*t^3+q^2*t^4+q^4*t+2*q^3*t^2+2*q^2*t^3\ +q*t^4+q^2*t^2+q*t^3)*s[2, 2, 1, 1]\ + (q^12+q^11*t+q^10*t+q^9*t^2+q^10+2*q^9*t+q^8*t^2+q^9+3*q^8*t+3*q^7*t^2+q^6*t^3+q^8+3*q^7*t\ +3*q^6*t^2+q^5*t^3+2*q^6*t+4*q^5*t^2+2*q^4*t^3+q^6+2*q^5*t+2*q^4*t^2+2*q^3*t^3+q^2*t^4\ +q^4*t+2*q^3*t^2+q^2*t^3)*s[2, 2, 2]\ + (q^12+q^11*t+q^11+2*q^10*t+q^9*t^2+2*q^10+4*q^9*t+2*q^8*t^2+2*q^9+5*q^8*t+4*q^7*t^2+q^6*t^3\ +2*q^8+6*q^7*t+5*q^6*t^2+q^5*t^3+q^7+5*q^6*t+6*q^5*t^2+2*q^4*t^3+q^6+4*q^5*t+5*q^4*t^2\ +2*q^3*t^3+2*q^4*t+4*q^3*t^2+2*q^2*t^3+q^3*t+2*q^2*t^2+q*t^3+q*t^2+t^3)*s[3, 1, 1, 1]\ + (q^11+q^10*t+2*q^10+3*q^9*t+q^8*t^2+2*q^9+5*q^8*t+3*q^7*t^2+3*q^8+7*q^7*t+5*q^6*t^2+q^5*t^3\ +3*q^7+8*q^6*t+7*q^5*t^2+2*q^4*t^3+2*q^6+7*q^5*t+7*q^4*t^2+2*q^3*t^3+2*q^5+5*q^4*t\ +5*q^3*t^2+2*q^2*t^3+q^4+3*q^3*t+3*q^2*t^2+q*t^3+q^2*t+q*t^2)*s[3, 2, 1]\ + (q^9+q^8*t+q^7*t+q^6*t^2+q^7+2*q^6*t+q^5*t^2+q^6+2*q^5*t+2*q^4*t^2+q^3*t^3+q^5+2*q^4*t\ +q^3*t^2+q^3*t+q^2*t^2+q^3+q^2*t)*s[3, 3]\ + (q^9+q^8*t+q^8+2*q^7*t+q^6*t^2+2*q^7+3*q^6*t+q^5*t^2+2*q^6+4*q^5*t+2*q^4*t^2+2*q^5+4*q^4*t\ +2*q^3*t^2+q^4+3*q^3*t+2*q^2*t^2+q^3+2*q^2*t+q*t^2+q*t+t^2)*s[4, 1, 1]\ + (q^8+q^7*t+q^7+2*q^6*t+q^5*t^2+2*q^6+3*q^5*t+q^4*t^2+q^5+3*q^4*t+2*q^3*t^2+2*q^4+3*q^3*t\ +q^2*t^2+q^3+2*q^2*t+q*t^2+q^2+q*t)*s[4, 2]\ + (q^5+q^4*t+q^4+q^3*t+q^3+q^2*t+q^2+q*t+q+t)*s[5, 1] + s[6] # Here are the coefficients for which we don't have a conjecture: # [2, 2] q * (q + t) * (q^2 + t + 1) # [3, 2] q * (q + t) * (q^4 + q^3 + q^2*t + q^2 + q*t + q + t + 1) # [2, 2, 1] q * (q + t) * (q^2 + t) * (q^4 + q^3 + q^2 + q*t + q + t + 1) # [2, 2, 2] (q + t) * q^2 * (q^2 + t) * (q^7 + q^5 + q^4*t + q^4 + q^3*t + q^3 + q^2*t + q*t + t^2 + q + t) # [3, 3] (q + t) * q^2 * (q^6 + q^4*t + q^4 + q^3*t + q^3 + q^2*t + q*t^2 + q^2 + q*t + t + 1) # [3, 2, 1] q * (q + t) * (q + 1)^2 * (q^2 + 1) * (q^2 + t) * (q^3 + t + 1) # [2, 2, 1, 1] q * (q + t) * (q^2 + t) * (q^2 + q + 1) * (q^3 + t) * (q^4 + q^2 + t + 1) # [4, 2] q * (q + t) * (q^2 + q + 1) * (q^4 + q^2*t + q^2 + t + 1)