A hereditary theorem for rigid and pseudorigid components of Lusztig's nilpotent varieties in type $A$

For irreducible components $C_σ$ of Lusztig's nilpotent varieties of type $A$ with graded dimension $(1,2,\dots,n,n-1,\dots,1)$ arising from permutations $σ\in S_n$, we characterize the property that $C_σ\oplus C_σ$ is an irreducible component combinatorially in terms of $σ$. The permutations that o