91Ó°ÊÓ

Use Theorem \(14.2 .3\) to determine the number of nonequivalent colorings of the corners of an equilateral triangle with the colors red and blue. Do the same with \(p\) colors (cf. Exercise 3).

How many different necklaces are there that contain three red and two blue beads?

How many different necklaces are there that contain four red and three blue beads?

Let \(f\) be a permutation of a set \(X\). Give a simple algorithm for finding the cycle factorization of \(f^{-1}\) from the cycle factorization of \(f\).

Show that the number of nonequivalent colorings of the corners of a regular 5 -gon with \(p\) colors is $$ \frac{p\left(p^{2}+4\right)\left(p^{2}+1\right)}{10} $$

Let \(n\) be an odd prime number. Prove that each of the permutations, \(\rho_{n}, \rho_{n}^{2}, \ldots, \rho_{n}^{n}\) of \(\\{1,2, \ldots, n\\}\) is an \(n\) -cycle. (Recall that \(\rho_{n}\) is the permutation that sends 1 to 2, 2 to \(3, \ldots, n-1\) to \(n\), and \(n\) to \(1 .\) )

Determine the cycle index of the dihedral group \(D_{2 p}\), where \(p\) is a prime number.

Find the generating function for the different necklaces that can be made with \(2 p\) beads each of color red or blue if \(p\) is a prime number.

Ten balls are stacked in a triangular array with 1 atop 2 atop 3 atop 4. (Think of billiards.) The triangular array is free to rotate. Find the generating function for the number of nonequivalent colorings with the colors red and blue. Find the generating function if we are also allowed to turn over the array.