A relation on 132-avoiding permutation patterns

Aisbett, Natalie

Discrete mathematics & theoretical computer science, Tome 17 (2015-2016) no. 2.

Résumé

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 $A$_$n$($τ$), 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.

Détail

DOI : 10.46298/dmtcs.2141