5 problems

Conjectures. For $E=E[\overline{123}]$ and for $E=E[\overline{213}]$, respectively,

Burstein–Jones's dominance conjecture. For all $n\geq0$,

Wilf-equivalence conjecture. The following Wilf-equivalences hold:

For consecutive permutation patterns, write $\sigma\stackrel{fs}{\sim}\tau$ when $\sigma$ and $\tau$ are filling-shape-Wilf-equivalent, meaning that they are avoided by fillings of ev…

A vincular pattern is a permutation pattern in which selected adjacent entries are required to occur consecutively; write $\sigma\sim\tau$ when two patterns are Wilf-equivalent, mea…