Irreducible characters with equal roots in the groups $S_n$ and $A_n$

We show that treating of (non-trivial) pairs of irreducible characters of the group $S_n$ sharing the same set of roots on one of the sets $A_n$ and $S_n\setminus A_n$ is divided into three parts. This, in particular, implies that any pair of such characters $\chi^\alpha$ and $\chi^\beta$ ($\alpha$ and $\beta$ are respective partitions of a number $n$) possesses the following property: lengths $d(\alpha)$ and $d(\beta)$ of principal diagonals of Young diagrams for $\alpha$ and $\beta$ differ by at most 1.