Aplikasi Teorema Polya dalam Menentukan Banyaknya Cara Pewarnaan Permukaan Oktahedron dengan m-Warna

One application of the permutation group concept is related to solving enumeration problems and one method that can solve this problem is Polya's Theorem. Polya's theorem is a calculation technique that combines abstract algebraic structures with combinatorics and can be used to calculate objects in permutation groups. A regular octahedron is an octahedron composed of eight equilateral triangles and the four sides of the octahedron meet at each vertex and have twelve edges. In 2015, research was carried out on the many ways to color the surface of a cube with m-colors. This research aims to determine the number of ways to color the surface of an octahedron with m-colors by using the Polya theorem and permutation groups to determine the rotational symmetry group formed by the octahedron as well as the number of cycle indices formed from each element in the rotational symmetry group itself. Based on the research results, it is obtained that the rotational symmetry group formed by the octahedron is S_4 and the number of cycle indices of the octahedron permutation group is Z(G) = 1/24*(x_1^8+6x_4^2+9x_2^4+8x_1^2*x_3^2) . After substituting the symmetry index and cycle index into the Polya theorem formula, we can conclude that the number of ways to color the octahedron surface with m-colors is 1/24*(m^8+17m^4+6m^2).