False
False
(True, None)
4

6

False

1 + 3*x + 9*x^2 + 27*x^3 + 81*x^4 + Order(x^5)

81

Order(1)

0
266
(x + 1)^2

y

3*x^6 + 3*x^4*y^2 + 2*x^3*y^3 + 3*x^2*y^4 + 3*y^6 + 2*x^5 + 2*x^3*y^2 + 2*x^2*y^3 + 2*y^5 + x^4 + 2*x^2*y^2 + y^4 + x^3 + y^3 + x^2 + y^2 + 1

756128164919912192

[(1/4*(-1)^(1/5)*(sqrt(5) + I*sqrt(2*sqrt(5) + 10) - 1), 1), (-1/4*(-1)^(1/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1), 1), (-1/4*(-1)^(1/5)*(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + 1), 1), (1/4*(-1)^(1/5)*(sqrt(5) - I*sqrt(2*sqrt(5) + 10) - 1), 1), ((-1)^(1/5), 1)]
[-0.309016994374947 + 0.951056516295154*I, -1.00000000000000, -0.309016994374947 - 0.951056516295154*I, 0.809016994374947 - 0.587785252292473*I, 0.809016994374947 + 0.587785252292473*I]

16.0000000000000 - 1.77635683940025e-15*I

()
(5,6)
(6,7)
(1,2)
(5,6,7)
(5,7,6)
(1,2)(6,7)
(1,2)(5,6)
(1,2)(5,6,7)
(1,2)(5,7,6)
(5,7)
(1,2)(5,7)
1/12*p[1, 1, 1, 1, 1, 1, 1] + 1/3*p[2, 1, 1, 1, 1, 1] + 1/4*p[2, 2, 1, 1, 1] + 1/6*p[3, 1, 1, 1, 1] + 1/6*p[3, 2, 1, 1]

The Group of Hexagon Symmetries has Polya cycle index:
1/12*p[1, 1, 1, 1, 1, 1] + 1/4*p[2, 2, 1, 1] + 1/3*p[2, 2, 2] + 1/6*p[3, 3] + 1/6*p[6]
(x1, x2, x3, x4, x6, r, g, b)
The number of hexagonal necklaces with red and blue beads has generating function:
b^6 + b^5*r + 3*b^4*r^2 + 3*b^3*r^3 + 3*b^2*r^4 + b*r^5 + r^6

()
(1,2,3,4,5)
(2,5)(3,4)
(1,2)(3,5)
(1,3,5,2,4)
(1,5)(2,4)
(1,3)(4,5)
(1,4)(2,3)
(1,4,2,5,3)
(1,5,4,3,2)
Notice the graph-theoretic symmetries of the 5-cycle graph are the same as the pentagon! Same group!
()
(1,5)(2,4)
(1,2,3,4,5)
(1,4)(2,3)
(1,3,5,2,4)
(2,5)(3,4)
(1,3)(4,5)
(1,5,4,3,2)
(1,4,2,5,3)
(1,2)(3,5)