M2
restart

-- three-cycle with all nodes the same color
-- all colorings up to symmetry
-- variance of the nodes normalized to be 1

n = 3

-- colors  colorings
--   1         1
--   2         1
--   3         1

-- using y for the original linear space, x for its inverse, l for colors

R  = QQ[l1, l2, l3, x11, x12, x13, x22, x23, x33]
X  = matrix{{x11, x12, x13},
            {x12, x22, x23},
            {x13, x23, x33}}

ID = matrix{{1,0,0},{0,1,0},{0,0,1}}

Y1 = matrix{{ 1, l1, l1},
            {l1,  1, l1},
            {l1, l1,  1}}

Y2 = matrix{{ 1, l1, l2},
            {l1,  1, l1},
            {l2, l1,  1}}

Y3 = matrix{{ 1, l1, l2},
            {l1,  1, l3},
            {l2, l3,  1}}

I1 = minors(1, X*Y1 - ID)
I2 = minors(1, X*Y2 - ID)
I3 = minors(1, X*Y3 - ID)

J1 = eliminate({l1,l2,l3},I1)
J2 = eliminate({l1,l2,l3},I2)
J3 = eliminate({l1,l2,l3},I3)

hasDegOne = p -> ((degree p)_0 == 1)  -- the _0 is because degree returns a list

LinForms1 = select(J1_*, hasDegOne)  -- the _* just lists all generators
LinForms2 = select(J2_*, hasDegOne)
LinForms3 = select(J3_*, hasDegOne)

print LinForms1
print LinForms2
print LinForms3

-----------------------------------------------------------------

restart

-- four-cycle with all nodes the same color
-- all colorings up to symmetry
-- numbering clockwise starting from upper left corner

-- 1--2
-- |  |
-- 4--3

n = 4

-- colors  partitions colorings
--   1     4              1
--   2     1+3, 2+2      1+2
--   3     1+1+2          2
--   4     1+1+1+1        1

-- using y for the original linear space, x for its inverse, l for colors

R  = QQ[l0, l1, l2, l3, l4, x11, x12, x13, x14, x22, x23, x24, x33, x34, x44]
X  = matrix{{x11, x12, x13, x14},
            {x12, x22, x23, x24},
            {x13, x23, x33, x34},
            {x14, x24, x34, x44}}

ID = matrix{{1,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1}}

Y1 = matrix{{l0, l1,  0, l1},
            {l1, l0, l1,  0},
            { 0, l1, l0, l1},
            {l1,  0, l1, l0}}

Y21 = matrix{{l0, l1,  0, l1},
             {l1, l0, l2,  0},
             { 0, l2, l0, l2},
             {l1,  0, l2, l0}}

Y22 = matrix{{l0, l1,  0, l2},
             {l1, l0, l2,  0},
             { 0, l2, l0, l1},
             {l2,  0, l1, l0}}

Y23 = matrix{{l0, l1,  0, l2},
             {l1, l0, l2,  0},
             { 0, l2, l0, l2},
             {l2,  0, l2, l0}}

Y31 = matrix{{l0, l1,  0, l1},
             {l1, l0, l2,  0},
             { 0, l2, l0, l3},
             {l1,  0, l3, l0}}

Y32 = matrix{{l0, l1,  0, l2},
             {l1, l0, l3,  0},
             { 0, l3, l0, l1},
             {l1,  0, l1, l0}}

Y4 = matrix{{l0, l1,  0, l2},
            {l1, l0, l3,  0},
            { 0, l3, l0, l4},
            {l2,  0, l4, l0}}

I1 = minors(1, X*Y1 - ID)
I21 = minors(1, X*Y21 - ID)
I22 = minors(1, X*Y22 - ID)
I23 = minors(1, X*Y23 - ID)
I31 = minors(1, X*Y31 - ID)
I32 = minors(1, X*Y32 - ID)
I4 = minors(1, X*Y4 - ID)

J1 = eliminate({l1,l2,l3,l4,l0}, I1)
J21 = eliminate({l1,l2,l3,l4,l0},I21)
J22 = eliminate({l1,l2,l3,l4,l0},I22)
J23 = eliminate({l1,l2,l3,l4,l0},I23)
J31 = eliminate({l1,l2,l3,l4,l0},I31)
J32 = eliminate({l1,l2,l3,l4,l0},I32)
J4 = eliminate({l1,l2,l3,l4,l0}, I4)

hasDegOne = p -> ((degree p)_0 == 1)  -- the _0 is because degree returns a list

LinForms1 = select(J1_*, hasDegOne)
LinForms21 = select(J21_*, hasDegOne)
LinForms22 = select(J22_*, hasDegOne)
LinForms23 = select(J23_*, hasDegOne)
LinForms31 = select(J31_*, hasDegOne)
LinForms32 = select(J32_*, hasDegOne)
LinForms4 = select(J4_*, hasDegOne)

print J1
print J21
print J22
print J23
print J31
print J32
print J4

-------------------------------------------------------------------

restart

-- five-cycle with all nodes the same color
-- all colorings up to symmetry

-- lay pentagon flat: 1--2--3--4--5--1

n = 5

-- partition    colorings

--   5            11111
--   4+1          11112
--   3+2          11122 11212
--   3+1+1        11123 11213
--   2+2+1        11223 11232 12123
--   2+1+1+1      11234 12134
--   1+1+1+1+1    12345

-- using y for the original linear space, x for its inverse, l for colors

R  = QQ[l0, l1, l2, l3, l4, l5, x11, x12, x13, x14, x15, x22, x23, x24, x25, x33, x34, x35, x44, x45, x55]
X  = matrix{{x11, x12, x13, x14, x15},
            {x12, x22, x23, x24, x25},
            {x13, x23, x33, x34, x35},
            {x14, x24, x34, x44, x45},
            {x15, x25, x35, x45, x55}}

ID = matrix{{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,0},{0,0,0,1,0},{0,0,0,0,1}}

Y1 = matrix{{l0, l1,  0,  0, l1},
            {l1, l0, l1,  0,  0},
            { 0, l1, l0, l1,  0},
            { 0,  0, l1, l0, l1},
            {l1,  0,  0, l1, l0}}

Y2 = matrix{{l0, l1,  0,  0, l2},
            {l1, l0, l1,  0,  0},
            { 0, l1, l0, l1,  0},
            { 0,  0, l1, l0, l1},
            {l2,  0,  0, l1, l0}}

Y31 = matrix{{l0, l1,  0,  0, l2},
            {l1, l0, l1,  0,  0},
            { 0, l1, l0, l1,  0},
            { 0,  0, l1, l0, l2},
            {l2,  0,  0, l2, l0}}

Y32 = matrix{{l0, l1,  0,  0, l2},
            {l1, l0, l1,  0,  0},
            { 0, l1, l0, l2,  0},
            { 0,  0, l2, l0, l1},
            {l2,  0,  0, l1, l0}}

Y41 = matrix{{l0, l1,  0,  0, l3},
            {l1, l0, l1,  0,  0},
            { 0, l1, l0, l1,  0},
            { 0,  0, l1, l0, l2},
            {l3,  0,  0, l2, l0}}

Y42 = matrix{{l0, l1,  0,  0, l3},
            {l1, l0, l1,  0,  0},
            { 0, l1, l0, l2,  0},
            { 0,  0, l2, l0, l1},
            {l3,  0,  0, l1, l0}}

Y51 = matrix{{l0, l1,  0,  0, l3},
            {l1, l0, l1,  0,  0},
            { 0, l1, l0, l2,  0},
            { 0,  0, l2, l0, l2},
            {l3,  0,  0, l2, l0}}

Y52 = matrix{{l0, l1,  0,  0, l2},
            {l1, l0, l1,  0,  0},
            { 0, l1, l0, l2,  0},
            { 0,  0, l2, l0, l3},
            {l2,  0,  0, l3, l0}}

Y53 = matrix{{l0, l1,  0,  0, l3},
            {l1, l0, l2,  0,  0},
            { 0, l2, l0, l1,  0},
            { 0,  0, l1, l0, l2},
            {l3,  0,  0, l2, l0}}

Y61 = matrix{{l0, l1,  0,  0, l4},
            {l1, l0, l1,  0,  0},
            { 0, l1, l0, l2,  0},
            { 0,  0, l2, l0, l3},
            {l4,  0,  0, l3, l0}}

Y62 = matrix{{l0, l1, 0, 0, l4},
            {l1, l0, l2,  0,  0},
            { 0, l2, l0, l1,  0},
            { 0,  0, l1, l0, l3},
            {l4,  0,  0, l3, l0}}

Y7 = matrix{{l0, l1,  0,  0, l2},
            {l1, l0, l3,  0,  0},
            { 0, l3, l0, l4,  0},
            { 0,  0, l4, l0, l5},
            {l2,  0,  0, l5, l0}}

models = {Y1, Y2, Y31, Y32, Y41, Y42, Y51, Y52, Y53, Y61, Y62, Y7}

modelIdeals = apply(models, p -> minors(1, X*p - ID))

eliminatedIdeals = apply(modelIdeals, p -> eliminate({l0,l1,l2,l3,l4,l5}, p))

hasDegOne = p -> ((degree p)_0 == 1)  -- the _0 is because degree returns a list

linForms = apply(eliminatedIdeals, p -> select(p_*, hasDegOne))

for p in linForms do print p