A relation on 132-avoiding permutation patterns

TL;DR

The paper investigates a conjecture relating permutation pattern avoidance counts to a structural order of associated trees, proving one direction and disproving the other with counterexamples.

Contribution

It proves one implication of Rudolph's conjecture and provides counterexamples to disprove the full conjecture about permutation pattern counts and tree structures.

Findings

Proved that certain tree structure orderings imply pattern avoidance count inequalities.

Disproved the conjecture's converse with explicit counterexamples.

Abstract

Rudolph conjectures that for permutations $p$ and $q$ of the same length, $A_n(p) \le A_n(q)$ 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 $A_n(p) \le A_n(q)$ for all $n$. We disprove the opposite direction by giving a counterexample, and hence disprove the conjecture.