A partially ordered pattern of length 
is a partially ordered
set on position labels 
specifying relative-value constraints for occurrences
in a permutation. An occurrence in 
consists of positions 
such that 
whenever 
. Incomparable labels impose no constraint (Kitaev 2007).
For example, the relations 
and 
, with 1 and 3 incomparable, describe a three-term subsequence
whose middle term is largest. Its classical permutation
patterns are 132 and 231. Avoiding this partially ordered pattern is equivalent
to avoiding both of those classical patterns.
In general, the corresponding classical patterns are the inverses of the linear extensions of the labeled partially ordered set. A total order therefore recovers a single classical permutation pattern. Biswas et al. (2026) study avoidance under combinations of these orders, including placing every element of one below every element of another. They classify the patterns of sizes 3, 4, and 5 whose components are chains.
