Egge triples and unbalanced Wilf equivalence

Egge, in a talk at the AMS Fall Eastern Meeting, 2012, conjectured that permutations avoiding the set of patterns {2143, 3142, τ}, where τ ∈ {246135, 254613, 524361, 546132, 263514}, are enumerated by the large Schröder numbers (and thus {2143, 3142, τ} with τ as above is Wilfequivalent to the set of patterns {2413, 3142}). Burstein and Pantone [J. Combin. 6 (2015) no. 1-2, 55–67] proved the case of τ = 246135. We prove the remaining four cases. As a byproduct of our proof, we also enumerate the case τ = 4132.