  Partially ordered patterns (POPs) generalize the notion of classical patterns
studied widely in the literature in the context of permutations, words,
compositions and partitions. In an occurrence of a POP, the relative order of
some of the elements is not important. Thus, any POP of length $k$ is defined
by a partially ordered set on $k$ elements, and classical patterns correspond
to $k$-element chains. The notion of a POP provides a convenient language to
deal with larger sets of permutation patterns.
  This paper contributes to a long line of research on classical permutation
patterns of length 4 and 5, and beyond, by conducting a systematic search of
connections between sequences in the Online Encyclopedia of Integer Sequences
(OEIS) and permutations avoiding POPs of length 4 and 5. As the result, we (i)
obtain 13 new enumerative results for classical patterns of length 4 and 5, and
a number of results for patterns of arbitrary length, (ii) collect under one
roof many sporadic results in the literature related to avoidance of patterns
of length 4 and 5, and (iii) conjecture 6 connections to the OEIS. Among the
most intriguing bijective questions we state, 7 are related to explaining
Wilf-equivalence of various sets of patterns, e.g.\ 5 or 8 patterns of length
4, and 2 or 6 patterns of length~5.
