Cycle Index Formulas for Dn Acting on Ordered Pairs

The cycle index of dihedral group D_n acting on the set X of the vertices of a regular n-gon was studied by Harary and Palmer in 1973 [1]. Since then a number of researchers have studied the cycle indices dihedral group acting on different sets and the resulting formulas have found applications in enumeration of a number of items. Muthoka (2015) [2] studied the cycle index formula of the dihedral group acting on unordered pairs from the set X={1, 2, , n} the n vertices of a regular n-gon. In this paper we study the cycle index formulas of D_n acting on ordered pairs from the set X={1, 2, , n}. In each case the actions of the cyclic part and the reflection part are studied separately for both an even value of n and an odd value of n.