A relation on 132-avoiding permutation patterns

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Rudolph conjectures that for permutations p and q of the same length, Aₙ(p) ≤ Aₙ(q) for all n if and only if the spine structure of T(p) is less than or equal to the spine structure of T(q) in refinement order. We prove one direction of this conjecture, by showing that if the spine structure of T(p) is less than or equal to the spine structure of T(q), then Aₙ(p) ≤ Aₙ(q) for all n. We disprove the opposite direction by giving a counterexample, and hence disprove the conjecture.