A permutation $σ$ contains the permutation $τ$ if there is a subsequence of $σ$ order isomorphic to $τ$. A permutation $σ$ is $τ$-<i>avoiding</i> if it does not contain the permutation $τ$. For any $n$, the <i>popularity</i> of a permutation $τ$, denoted $A$<sub>$n$</sub>($τ$), is the number of copies of $τ$ contained in the set of all 132-avoiding permutations of length $n$. Rudolph conjectures that for permutations $τ$ and $μ$ of the same length, $A$<sub>$n$</sub>($τ$) ≤ $A$<sub>$n$</sub>($μ$) for all $n$ if and only if the spine structure of $τ$ is less than or equal to the spine structure of $μ$ in refinement order. We prove one direction of this conjecture, by showing that if the spine structure of $τ$ is less than or equal to the spine structure of $μ$, then $A$<sub>$n$</sub>($τ$) ≤ $A$<sub>$n$</sub>($μ$) for all $n$. We disprove the opposite direction by giving a counterexample, and hence disprove the conjecture.
Discrete Mathematics & Theoretical Computer Science