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Discrete Mathematics & Theoretical Computer Science |
Natalie Aisbett - A relation on 132-avoiding permutation patterns
dmtcs:2141 - Discrete Mathematics & Theoretical Computer Science, December 15, 2015, Vol. 17 no.2 - https://doi.org/10.46298/dmtcs.2141- 1 School of Mathematics and statistics [Sydney]
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.