• 1 School of Mathematics and statistics [Sydney]

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