A new inequality related to the Diaconis-Graham inequalities and a new characterisation of the dihedral group

We prove an inequality between three measures of disorder on the symmetric group on n elements. This inequality has been inspired by the well-known Diaconis-Graham inequalities. We also discuss when the inequality is satisfied as equality, and how often this happens. In the case n is odd, the number of permutations that satisfy the equality is a simple function of the Lucas numbers. In addition, we show that a quantity involved in the new inequality (which is a function of the three measures of disorder) can be used to give a new characterization of the dihedral group.