I hope I understood the question correctly. I have a feeling that questions on permutations of algebraic as opposed to combinatorial nature, could be candidates.

Lakshmibai and Sandhya's theorem is a geometric question and it is a significant theorem because it reduces geometry to combinatorics. With this understanding of your question let me attempt to give four examples:

(1) A permutation being of specific order $m$ .

Suppose we attempt pattern avoidance like: for any $k$ relatively prime to $m$ it should not have a length $k$ cycle. A permutation of order, for example $m^2$, will also satisfy that criterion and will be accepted wrongly.

(2) Permutation being even. (avoidance criterion may not work: because presence of an even number of cycles of any particular length, as opposed odd number of them, will be ok)

(3) Some irreducible character vanishing in it. This is conjugacy class question. Can be argued similarly

(4) Commuting with another specific permutation.