SharedHome directory / simplices.sagewsOpen in CoCalc
f=0
i=0
g=0
aa=0
p = SymmetricFunctions(QQ).power()
t=0
for G in graphs(6):
        if G.is_connected():
            Aut=G.automorphism_group()
            E=G.size()
            XG = G.chromatic_symmetric_function();
            i=i+1
            f=f+p(XG)*(1/Aut.order())
            
            aa=aa+(1/Aut.order())
            
f
1669/45*p[1, 1, 1, 1, 1, 1] - 595/2*p[2, 1, 1, 1, 1] + 3807/8*p[2, 2, 1, 1] - 675/8*p[2, 2, 2] + 9275/18*p[3, 1, 1, 1] - 1645/2*p[3, 2, 1] + 12775/72*p[3, 3] - 790*p[4, 1, 1] + 6715/16*p[4, 2] + 104687/120*p[5, 1] - 362431/720*p[6]
f=0
i=0
g=0
aa=0
p = SymmetricFunctions(QQ).power()
t=0
for G in graphs(5):
        if G.is_connected():
            Aut=G.automorphism_group()
            E=G.size()
            XG = G.chromatic_symmetric_function();
            i=i+1
            f=f+p(XG)*(1/Aut.order())
            #g=g+p(XG)*(1/Aut.order())
            aa=aa+(1/Aut.order())
f          
91/15*p[1, 1, 1, 1, 1] - 69/2*p[2, 1, 1, 1] + 117/4*p[2, 2, 1] + 140/3*p[3, 1, 1] - 105/4*p[3, 2] - 395/8*p[4, 1] + 3377/120*p[5]
f=0
i=0
g=0
aa=0
p = SymmetricFunctions(QQ).power()
t=0
for G in graphs(4):
        if G.is_connected():
            Aut=G.automorphism_group()
            E=G.size()
            XG = G.chromatic_symmetric_function();
            i=i+1
            f=f+p(XG)*(1/Aut.order())
            #g=g+p(XG)*(1/Aut.order())
            aa=aa+(1/Aut.order())
f
19/12*p[1, 1, 1, 1] - 6*p[2, 1, 1] + 15/8*p[2, 2] + 35/6*p[3, 1] - 79/24*p[4]
f=0
i=0
g=0
aa=0
p = SymmetricFunctions(QQ).power()
t=0
for G in graphs(3):
        if G.is_connected():
            Aut=G.automorphism_group()
            E=G.size()
            XG = G.chromatic_symmetric_function();
            i=i+1
            f=f+p(XG)*(1/Aut.order())
            #g=g+p(XG)*(1/Aut.order())
            aa=aa+(1/Aut.order())
f
2/3*p[1, 1, 1] - 3/2*p[2, 1] + 5/6*p[3]

f=0
i=0
g=0
aa=0
p = SymmetricFunctions(QQ).power()
t=0
for G in graphs(2):
        if G.is_connected():
            Aut=G.automorphism_group()
            E=G.size()
            XG = G.chromatic_symmetric_function();
            i=i+1
            f=f+p(XG)*(1/Aut.order())
            #g=g+p(XG)*(1/Aut.order())
            aa=aa+(1/Aut.order())
f
1/2*p[1, 1] - 1/2*p[2]
Gr=1+p[1]+1/2*p[1, 1] - 1/2*p[2]+2/3*p[1, 1, 1] - 3/2*p[2, 1] + 5/6*p[3]+19/12*p[1, 1, 1, 1] - 6*p[2, 1, 1] + 15/8*p[2, 2] + 35/6*p[3, 1] - 79/24*p[4]+91/15*p[1, 1, 1, 1, 1] - 69/2*p[2, 1, 1, 1] + 117/4*p[2, 2, 1] + 140/3*p[3, 1, 1] - 105/4*p[3, 2] - 395/8*p[4, 1] + 3377/120*p[5]+1669/45*p[1, 1, 1, 1, 1, 1] - 595/2*p[2, 1, 1, 1, 1] + 3807/8*p[2, 2, 1, 1] - 675/8*p[2, 2, 2] + 9275/18*p[3, 1, 1, 1] - 1645/2*p[3, 2, 1] + 12775/72*p[3, 3] - 790*p[4, 1, 1] + 6715/16*p[4, 2] + 104687/120*p[5, 1] - 362431/720*p[6]
M0=matrix([[2*15/8,35/6,4*3*2*19/12,1],[2*117/4,2*140/3,5*4*3*2*91/15,2*3*2*2/3],[2*3807/8,3*9275/18,6*5*4*3*1669/45,(3*2*2/3)**2+2*4*3*19/12],[3*2*675/8,1645/2,595/2*4*3*2,2*12]])
M0.rank()
3

f=0
i=0
g=0
aa=0
p = SymmetricFunctions(QQ).power()
t=0
for G in graphs(6):
        if G.is_connected():
            Aut=G.automorphism_group()
            E=G.size()
            XG = G.chromatic_symmetric_function();
            i=i+1
            f=f+p(XG)*(1/Aut.order())
            g=g+p(XG)*(1/Aut.order())
            aa=aa+(1/Aut.order())
            K=G.clique_complex()
            S5=Set(K.n_cells(5))
            U5=S5.subsets()
            for u5 in U5:
                L5=SimplicialComplex(Set(u5))
                S4=Set(K.n_cells(4,subcomplex=L5 ))
                U4=S4.subsets()
                for u4 in U4:
                    L4=SimplicialComplex(Set(u4)+Set(u5))
                    S3=Set(K.n_cells(3,subcomplex=L4))
                    U3=S3.subsets()
                    for u3 in U3:
                        #print(u)
                        #Q=S[len(S)-1]
                        L3=SimplicialComplex(Set(u3)+Set(u4)+Set(u5))
                        #print(L)
                        t=t+(2**(len(list(K.n_cells(2, subcomplex=L3)))))*1/Aut.order()*p(XG)
                   

print(t)
483983/45*p[1, 1, 1, 1, 1, 1] - 299559/2*p[2, 1, 1, 1, 1] + 3349239/8*p[2, 2, 1, 1] - 2086931/16*p[2, 2, 2] + 6921233/18*p[3, 1, 1, 1] - 4306941/4*p[3, 2, 1] + 2771853/8*p[3, 3] - 13357403/16*p[4, 1, 1] + 37463963/48*p[4, 2] + 77408939/60*p[5, 1] - 748272391/720*p[6] 483983/45*p[1, 1, 1, 1, 1, 1] - 299559/2*p[2, 1, 1, 1, 1] + 3349239/8*p[2, 2, 1, 1] - 2086931/16*p[2, 2, 2] + 6921233/18*p[3, 1, 1, 1] - 4306941/4*p[3, 2, 1] + 2771853/8*p[3, 3] - 13357403/16*p[4, 1, 1] + 37463963/48*p[4, 2] + 77408939/60*p[5, 1] - 748272391/720*p[6]
(483983/45*p[1, 1, 1, 1, 1, 1] - 299559/2*p[2, 1, 1, 1, 1] + 3349239/8*p[2, 2, 1, 1] - 2086931/16*p[2, 2, 2] + 6921233/18*p[3, 1, 1, 1] - 4306941/4*p[3, 2, 1] + 2771853/8*p[3, 3] - 13357403/16*p[4, 1, 1] + 37463963/48*p[4, 2] + 77408939/60*p[5, 1] - 748272391/720*p[6])
483983/45*p[1, 1, 1, 1, 1, 1] - 299559/2*p[2, 1, 1, 1, 1] + 3349239/8*p[2, 2, 1, 1] - 2086931/16*p[2, 2, 2] + 6921233/18*p[3, 1, 1, 1] - 4306941/4*p[3, 2, 1] + 2771853/8*p[3, 3] - 13357403/16*p[4, 1, 1] + 37463963/48*p[4, 2] + 77408939/60*p[5, 1] - 748272391/720*p[6]
f=0
i=0
g=0
aa=0
p = SymmetricFunctions(QQ).power()
t=0
for G in graphs(4):
        if G.is_connected():
            Aut=G.automorphism_group()
            E=G.size()
            XG = G.chromatic_symmetric_function();
            i=i+1
            f=f+p(XG)*(1/Aut.order())
            g=g+p(XG)*(1/Aut.order())
            aa=aa+(1/Aut.order())
            K=G.clique_complex()
            S5=Set(K.n_cells(5))
            U5=S5.subsets()
            for u5 in U5:
                L5=SimplicialComplex(Set(u5))
                S4=Set(K.n_cells(4,subcomplex=L5 ))
                U4=S4.subsets()
                for u4 in U4:
                    L4=SimplicialComplex(Set(u4)+Set(u5))
                    S3=Set(K.n_cells(3,subcomplex=L4))
                    U3=S3.subsets()
                    for u3 in U3:
                        #print(u)
                        #Q=S[len(S)-1]
                        L3=SimplicialComplex(Set(u3)+Set(u4)+Set(u5))
                        #print(L)
                        t=t+(2**(len(list(K.n_cells(2, subcomplex=L3)))))*1/Aut.order()*p(XG)
                    

print(t)
7/2*p[1, 1, 1, 1] - 63/4*p[2, 1, 1] + 47/8*p[2, 2] + 53/3*p[3, 1] - 271/24*p[4]

f=0
i=0
g=0
aa=0
p = SymmetricFunctions(QQ).power()
t=0
for G in graphs(5):
        if G.is_connected():
            Aut=G.automorphism_group()
            E=G.size()
            XG = G.chromatic_symmetric_function();
            i=i+1
            f=f+p(XG)*(1/Aut.order())
            g=g+p(XG)*(1/Aut.order())
            aa=aa+(1/Aut.order())
            K=G.clique_complex()
            S5=Set(K.n_cells(5))
            U5=S5.subsets()
            for u5 in U5:
                L5=SimplicialComplex(Set(u5))
                S4=Set(K.n_cells(4,subcomplex=L5 ))
                U4=S4.subsets()
                for u4 in U4:
                    L4=SimplicialComplex(Set(u4)+Set(u5))
                    S3=Set(K.n_cells(3,subcomplex=L4))
                    U3=S3.subsets()
                    for u3 in U3:
                        #print(u)
                        #Q=S[len(S)-1]
                        L3=SimplicialComplex(Set(u3)+Set(u4)+Set(u5))
                        #print(L)
                        t=t+(2**(len(list(K.n_cells(2, subcomplex=L3)))))*1/Aut.order()*p(XG)
                    

print(t)
529/10*p[1, 1, 1, 1, 1] - 1279/3*p[2, 1, 1, 1] + 4159/8*p[2, 2, 1] + 9083/12*p[3, 1, 1] - 2475/4*p[3, 2] - 6115/6*p[4, 1] + 88147/120*p[5]

f=0
i=0
g=0
aa=0
p = SymmetricFunctions(QQ).power()
t=0
for G in graphs(3):
        if G.is_connected():
            Aut=G.automorphism_group()
            E=G.size()
            XG = G.chromatic_symmetric_function();
            i=i+1
            f=f+p(XG)*(1/Aut.order())
            g=g+p(XG)*(1/Aut.order())
            aa=aa+(1/Aut.order())
            K=G.clique_complex()
            S5=Set(K.n_cells(5))
            U5=S5.subsets()
            for u5 in U5:
                L5=SimplicialComplex(Set(u5))
                S4=Set(K.n_cells(4,subcomplex=L5 ))
                U4=S4.subsets()
                for u4 in U4:
                    L4=SimplicialComplex(Set(u4)+Set(u5))
                    S3=Set(K.n_cells(3,subcomplex=L4))
                    U3=S3.subsets()
                    for u3 in U3:
                        #print(u)
                        #Q=S[len(S)-1]
                        L3=SimplicialComplex(Set(u3)+Set(u4)+Set(u5))
                        #print(L)
                        t=t+(2**(len(list(K.n_cells(2, subcomplex=L3)))))*1/Aut.order()*p(XG)
                   

print(t)
5/6*p[1, 1, 1] - 2*p[2, 1] + 7/6*p[3]
f=0
i=0
g=0
aa=0
p = SymmetricFunctions(QQ).power()
t=0
for G in graphs(2):
        if G.is_connected():
            Aut=G.automorphism_group()
            E=G.size()
            XG = G.chromatic_symmetric_function();
            i=i+1
            f=f+p(XG)*(1/Aut.order())
            g=g+p(XG)*(1/Aut.order())
            aa=aa+(1/Aut.order())
            K=G.clique_complex()
            S5=Set(K.n_cells(5))
            U5=S5.subsets()
            for u5 in U5:
                L5=SimplicialComplex(Set(u5))
                S4=Set(K.n_cells(4,subcomplex=L5 ))
                U4=S4.subsets()
                for u4 in U4:
                    L4=SimplicialComplex(Set(u4)+Set(u5))
                    S3=Set(K.n_cells(3,subcomplex=L4))
                    U3=S3.subsets()
                    for u3 in U3:
                        #print(u)
                        #Q=S[len(S)-1]
                        L3=SimplicialComplex(Set(u3)+Set(u4)+Set(u5))
                        #print(L)
                        t=t+(2**(len(list(K.n_cells(2, subcomplex=L3)))))*1/Aut.order()*p(XG)
                    

print(t)
1/2*p[1, 1] - 1/2*p[2]
F=1+p[1]+1/2*p[1, 1] + 1/2*p[2]+5/6*p[1, 1, 1] + 2*p[2, 1] + 7/6*p[3]+529/10*p[1, 1, 1, 1, 1] + 1279/3*p[2, 1, 1, 1] + 4159/8*p[2, 2, 1] + 9083/12*p[3, 1, 1] + 2475/4*p[3, 2]+6115/6*p[4, 1] + 88147/120*p[5]+483983/45*p[1, 1, 1, 1, 1, 1]+ 299559/2*p[2, 1, 1, 1, 1] + 3349239/8*p[2, 2, 1, 1] +2086931/16*p[2, 2, 2] + 6921233/18*p[3, 1, 1, 1]+ 4306941/4*p[3, 2, 1] + 2771853/8*p[3, 3] +13357403/16*p[4, 1, 1] + 37463963/48*p[4, 2] + 77408939/60*p[5, 1] + 748272391/720*p[6]+7/2*p[1, 1, 1, 1] + 63/4*p[2, 1, 1] + 47/8*p[2, 2] + 53/3*p[3, 1] +271/24*p[4]
p = SymmetricFunctions(QQ).p()
Fdp2dp2=  2*4159/8*p[ 1]  + 3349239/8*2*p[1, 1] +3*2*2086931/16*p[ 2] + 2*47/8
Fdp3dp1= 2*9083/12*p[1] + 3*6921233/18*p[1, 1]+ 4306941/4*p[ 2]+53/3
Fdp1_4=5*4*3*2*529/10*p[1]+6*5*4*3*483983/45*p[1,1]+4*3*2*271/24+4*3*2*299559/2*p[2]
Fdp1_2=1+3*2*5/6*p[1]+5*4*529/10*p[1, 1, 1] + 3*2*1279/3*p[2, 1] + 2*1* 9083/12*p[3] +6*5*483983/45*p[1, 1, 1, 1]+4*3* 299559/2*p[2, 1, 1] + 2*1*3349239/8*p[2, 2]  + 3*2*6921233/18*p[3, 1]+2*1*13357403/16*p[4] +4*3*7/2*p[1, 1] + 2*1*63/4*p[2] 
Fdp1_2**2
p[] + 10*p[1] + 109*p[1, 1] + 2536*p[1, 1, 1] + 1972964/3*p[1, 1, 1, 1] + 9946276/3*p[1, 1, 1, 1, 1] + 28222412*p[1, 1, 1, 1, 1, 1] + 2048216056/3*p[1, 1, 1, 1, 1, 1, 1] + 936958177156/9*p[1, 1, 1, 1, 1, 1, 1, 1] + 63*p[2] + 5431*p[2, 1] + 3622934*p[2, 1, 1] + 18255066*p[2, 1, 1, 1] + 176717750*p[2, 1, 1, 1, 1] + 16361717248/3*p[2, 1, 1, 1, 1, 1] + 1159851707976*p[2, 1, 1, 1, 1, 1, 1] + 6702447/4*p[2, 2] + 17068503/2*p[2, 2, 1] + 190110685*p[2, 2, 1, 1] + 10967010495*p[2, 2, 1, 1, 1] + 3770806314295*p[2, 2, 1, 1, 1, 1] + 211002057/4*p[2, 2, 2] + 4283676681*p[2, 2, 2, 1] + 3009884056803*p[2, 2, 2, 1, 1] + 11217401879121/16*p[2, 2, 2, 2] + 9083/3*p[3] + 13887881/3*p[3, 1] + 69593816/3*p[3, 1, 1] + 590993386/3*p[3, 1, 1, 1] + 52728022262/9*p[3, 1, 1, 1, 1] + 13399036444156/9*p[3, 1, 1, 1, 1, 1] + 190743/2*p[3, 2] + 459271993/3*p[3, 2, 1] + 51734394410/3*p[3, 2, 1, 1] + 8293270544988*p[3, 2, 1, 1, 1] + 10140379279/4*p[3, 2, 2] + 7726954497229/2*p[3, 2, 2, 1] + 82500889/36*p[3, 3] + 62865559339/9*p[3, 3, 1] + 47903466240289/9*p[3, 3, 1, 1] + 13357403/4*p[4] + 66787015/4*p[4, 1] + 280505463/2*p[4, 1, 1] + 7066066187/2*p[4, 1, 1, 1] + 6464755976149/6*p[4, 1, 1, 1, 1] + 841516389/8*p[4, 2] + 17084118437/2*p[4, 2, 1] + 12003990855831/2*p[4, 2, 1, 1] + 44737135066317/16*p[4, 2, 2] + 121325291449/24*p[4, 3] + 92449698437899/12*p[4, 3, 1] + 178420214904409/64*p[4, 4]
s = SymmetricFunctions(QQ['t'].fraction_field()).schur()


Eq0=A*2*47/8*c2^2+B*53/3*c3*c1+C*4*3*2*7/2*c1**4+D*1*c1**4
Eq1=A*2*4159/8*c2**2*c1+B*2*9083/12*c3*c1**2+C*5*4*3*2*529/10*c1**5+D*10*c1**5
Eq11=A*2*3349239/8*c2**2*c1**2+B*3*6921233/18*c3*c1**3+C*6*5*4*3*483983/45*c1**6+D*109*c1**6
Eq2=A*3*2*2086931/16*c2**3+B*4306941/4*c2*c3*c1+C*299559/2*4*3*2*c2*c1**4+D*(63*c2)*c1**4
A, B, C,D, c3, c2 = var('A, B, C,D, c3, c2')

Eq0=A*2*47/8*c2^2+B*53/3*c3*c1+C*4*3*2*7/2*c1**4+D*1*c1**4
Eq1=A*2*4159/8*c2**2*c1+B*2*9083/12*c3*c1**2+C*5*4*3*2*529/10*c1**5+D*10*c1**5
Eq11=A*2*3349239/8*c2**2*c1**2+B*3*6921233/18*c3*c1**3+C*6*5*4*3*483983/45*c1**6+D*109*c1**6
Eq2=A*3*2*2086931/16*c2**3+B*4306941/4*c2*c3*c1+C*299559/2*4*3*2*c2*c1**4+D*(63*c2)*c1**4
Eq0
47/4*A*c2^2 + 53/3*B*c3 + 84*C + D
F=1+p[1]+1/2*p[1, 1] + 1/2*p[2]+5/6*p[1, 1, 1] + 2*p[2, 1] + 7/6*p[3]+529/10*p[1, 1, 1, 1, 1] + 1279/3*p[2, 1, 1, 1] + 4159/8*p[2, 2, 1] + 9083/12*p[3, 1, 1] + 2475/4*p[3, 2]+6115/6*p[4, 1] + 88147/120*p[5]+483983/45*p[1, 1, 1, 1, 1, 1]+ 299559/2*p[2, 1, 1, 1, 1] + 3349239/8*p[2, 2, 1, 1] +2086931/16*p[2, 2, 2] + 6921233/18*p[3, 1, 1, 1]+ 4306941/4*p[3, 2, 1] + 2771853/8*p[3, 3] +13357403/16*p[4, 1, 1] + 37463963/48*p[4, 2] + 77408939/60*p[5, 1] + 748272391/720*p[6]+7/2*p[1, 1, 1, 1] + 63/4*p[2, 1, 1] + 47/8*p[2, 2] + 53/3*p[3, 1] +271/24*p[4]
M=matrix([[2*47/8,53/3,4*3*2*7/2,1],[2*4159/8,2*9083/12,5*4*3*2*529/10,10],[2*3349239/8,3*6921233/18,6*5*4*3*483983/45,109],[3*2*2086931/16,4306941/4,299559/2*4*3*2,63]])
M.rank()
M

4 [ 47/4 53/3 84 1] [ 4159/4 9083/6 6348 10] [3349239/4 6921233/6 3871864 109] [6260793/8 4306941/4 3594708 63]

from scipy import linalg
linalg.svd(M)[1]
array([ 5.63176978e+06, 5.37717522e+03, 9.83704518e+00, 4.90813610e-01])
 
solve([Eq0==0,Eq1==0, Eq11==0, Eq2==0],A, B, C,D,c2,c3)
[[A == r318, B == 0, C == 0, D == 0, c2 == 0, c3 == r319], [A == r320, B == r321, C == 0, D == 0, c2 == 0, c3 == 0], [A == 0, B == 0, C == 0, D == 0, c2 == r322, c3 == r323], [A == 0, B == r324, C == 0, D == 0, c2 == r325, c3 == 0]]
f=0
i=0
g=0
aa=0
p = SymmetricFunctions(QQ).power()
t=0
for G in graphs(7):
        if G.is_connected():
            Aut=G.automorphism_group()
            E=G.size()
            XG = G.chromatic_symmetric_function();
            i=i+1
            f=f+p(XG)*(1/Aut.order())
            g=g+p(XG)*(1/Aut.order())
            aa=aa+(1/Aut.order())
            K=G.clique_complex()
            S5=Set(K.n_cells(5))
            U5=S5.subsets()
            for u5 in U5:
                L5=SimplicialComplex(Set(u5))
                S4=Set(K.n_cells(4,subcomplex=L5 ))
                U4=S4.subsets()
                for u4 in U4:
                    L4=SimplicialComplex(Set(u4)+Set(u5))
                    S3=Set(K.n_cells(3,subcomplex=L4))
                    U3=S3.subsets()
                    for u3 in U3:
                        #print(u)
                        #Q=S[len(S)-1]
                        L3=SimplicialComplex(Set(u3)+Set(u4)+Set(u5))
                        #print(L)
                        t=t+(2**(len(list(K.n_cells(2, subcomplex=L3)))))*1/Aut.order()*p(XG)
                    #print(L3,len(list(K.n_cells(2, subcomplex=L3))))
                    #if len(list(K.n_cells(2, subcomplex=L)))<>20:
                            #print(len(list(K.n_faces(2, subcomplex=L))))

            #print("number",i)

print(t)
Error in lines 7-29 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1013, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 5, in <module> File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/graphs/graph.py", line 4394, in chromatic_symmetric_function ret += (-1)**len(F) * p[la] File "sage/structure/element.pyx", line 1239, in sage.structure.element.Element.__add__ (build/cythonized/sage/structure/element.c:10849) return (<Element>left)._add_(right) File "sage/modules/with_basis/indexed_element.pyx", line 590, in sage.modules.with_basis.indexed_element.IndexedFreeModuleElement._add_ (build/cythonized/sage/modules/with_basis/indexed_element.c:6985) add(self._monomial_coefficients, File "sage/data_structures/blas_dict.pyx", line 270, in sage.data_structures.blas_dict.add (build/cythonized/sage/data_structures/blas_dict.c:2483) return axpy(1, D2, D) File "sage/data_structures/blas_dict.pyx", line 203, in sage.data_structures.blas_dict.axpy (build/cythonized/sage/data_structures/blas_dict.c:1931) iaxpy(a, X, Y, True, factor_on_left) File "sage/data_structures/blas_dict.pyx", line 137, in sage.data_structures.blas_dict.iaxpy (build/cythonized/sage/data_structures/blas_dict.c:1569) if key in Y: File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/combinat/combinat.py", line 930, in __eq__ def __eq__(self, other): File "src/cysignals/signals.pyx", line 251, in cysignals.signals.python_check_interrupt File "src/cysignals/signals.pyx", line 94, in cysignals.signals.sig_raise_exception KeyboardInterrupt