The young diagrams of a pair of irreducible characters of $S_n$ with the same zero set on $S^\varepsilon_n$

In studying the pairs of irreducible characters of the symmetric group $S_n$ with the same zero set on $A_n$ or $S_n\setminus A_n$ (as well as the pairs of irreducible characters with the same zero set on the alternating group $A_n$), the results are important on the connection between the Young diagrams of the characters of these pairs. We prove a theorem that considerably generalizes two previous results of frequent use in this direction.