  We prove a conjecture of Gao and Kitaev on Wilf-equivalence of sets of
patterns {12345,12354} and {45123,45213} that extends the list of 10 related
conjectures proved in the literature in a series of papers. To achieve our
goals, we prove generalized versions of shape-Wilf-equivalence results of
Backelin, West, and Xin and use a particular result on shape-Wilf-equivalence
of monotone patterns. We also derive general results on shape-Wilf-equivalence
of certain classes of partially ordered patterns and use their specialization
(also appearing in a paper by Bloom and Elizalde) as an essential piece in
proving the conjecture. Our results allow us to show (shape-)Wilf-equivalence
of large classes of sets of patterns, including 11 out of 12 classes found by
Bean et al. in relation to the conjecture.
