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A relation on 132-avoiding permutation patterns

Abstract : A permutation $σ$ contains the permutation $τ$ if there is a subsequence of $σ$ order isomorphic to $τ$. A permutation $σ$ is $τ$-avoiding if it does not contain the permutation $τ$. For any $n$, the popularity of a permutation $τ$, denoted $A$$n$($τ$), is the number of copies of $τ$ contained in the set of all 132-avoiding permutations of length $n$. Rudolph conjectures that for permutations $τ$ and $μ$ of the same length, $A$$n$($τ$) ≤ $A$$n$($μ$) for all $n$ if and only if the spine structure of $τ$ is less than or equal to the spine structure of $μ$ in refinement order. We prove one direction of this conjecture, by showing that if the spine structure of $τ$ is less than or equal to the spine structure of $μ$, then $A$$n$($τ$) ≤ $A$$n$($μ$) for all $n$. We disprove the opposite direction by giving a counterexample, and hence disprove the conjecture.
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Natalie Aisbett. A relation on 132-avoiding permutation patterns. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2015, Vol. 17 no.2 (2), pp.285-302. ⟨hal-01349056⟩

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