Proof of a conjecture on the shape-Wilf-equivalence for partially ordered patterns

A partially ordered pattern (abbreviated POP) is a partially ordered set (poset) that generalizes the notion of a pattern when we are not concerned with the relative order of some of its letters. The notion of partially ordered patterns provides a convenient language to deal with large sets of permutation patterns. In analogy to the shape-Wilf-equivalence for permutation patterns, Burstein–Han–Kitaev–Zhang initiated the study of the shape-Wilf-equivalence for POPs which would result in the shape-Wilf-equivalence for large sets of permutation patterns. The main objective of this paper is to confirm a recent intriguing conjecture posed by Burstein–Han–Kitaev–Zhang concerning the shape-Wilf-equivalence for POPs of length $k$. This is accomplished by establishing a bijection between two sets of pattern-avoiding transversals of a given Young diagram.