A large minimal blocker for 123-avoiding permutations

Abstract

A set $B\subseteq \{1,\dots ,n\}\times \{1,\dots ,n\}$ is a blocker of a subset $Q_n$ of the symmetric group $S_n$ if every permutation $\sigma \in Q_n$ allows an index i with $(i,{\sigma }_i)\in B$. Bennett, Brualdi and Cao conjectured that $\lceil (n+1)/2\rceil \cdot \lfloor (n+1)/2\rfloor$ is an upper bound for the sizes of the inclusion minimal blockers of the family of 123-avoiding permutations, which are those $\sigma \in S_n$ for which $({\sigma }_1,\dots ,{\sigma }_n)$ has no increasing subsequence of the length three. We show that$B=\left(\begin{array}{ccccccc}0& 0& 0& 0& ⁎& 0& ⁎\\ 0& 0& ⁎& ⁎& 0& 0& ⁎\\ 0& 0& 0& 0& 0& 0& 0\\ 0& ⁎& 0& ⁎& ⁎& 0& ⁎\\ 0& 0& 0& 0& ⁎& 0& 0\\ 0& ⁎& 0& ⁎& ⁎& 0& ⁎\\ 0& ⁎& 0& ⁎& 0& 0& ⁎\end{array}\right)$ is a counterexample to the conjecture, where the ⁎'s denote the positions in the blocker.