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Unformatted text preview: C&O 330 - ASSIGNMENT #5 DUE FRIDAY, 19 DECEMBER AT 10:31PM (1) (20 points) (a) (15 points) Let u < and d . Let a 1 , . . . , a 4 be positive integers. Find the number of permutations with pattern u a 1- 1 du a 2- 1 du a 3- 1 du a 4- 1 , expressing your result as a determinant of binomial numbers. Solution: This part is a slight generalisation of the similar question on the previous assignment. It is done in the same way, and many of the details are in the Notes! The result is det ( M 4 ) where M 4 = bracketleftBigg parenleftbigg 5- i k =1 a k 5- i k =5- j a k parenrightbigg bracketrightBigg 4 4 (b) (5 points) On the basis of this evidence state a conjecture of the number of permutations with pattern u a 1- 1 du a 2- 1 d u a m- 1 du a m +1- 1 . Solution: The conjecture is det ( M m +1 ) where M m +1 = bracketleftBigg parenleftbigg m +2- i k =1 a k m +2- i k =5- j a k parenrightbigg bracketrightBigg ( m +1) ( m +1) . (2) (20 points) A dodecahedron is a regular solid consisting of 12 regular pentagons arranged so that each vertex of the dodecahedron is incident with 3 pentagons and that every edge of the dodecahedron is incident with 2 pentagons. (a) (15 points) Find the cycle index polynomial for the automorphism group of the dodecahedron acting on the 12 faces of the dodecahedron. Solution: The automorphisms come from rotations around axes spec- ified by a) two antipodal vertices, b) two andipodal edges and c) two faces. Case (a): There are 20 vertices, and therefore 10 pairs of antipodal vertices and therefore 10 axes of rotation passing through antipodal vertices..If c is a rotation through 120 degrees, then c and c 2 are au- tomorphisms. Thus there are 20 automorphisms, each consisting of 4 1 2 DUE FRIDAY, 19 DECEMBER AT 10:31PM 3-cycles. The contribution of these to the cycle index polynomial is3-cycles....
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- Spring '05