# On partially ordered patterns of length 4 and 5 in permutations

## TL;DR

This paper systematically explores the enumeration of permutations avoiding partially ordered patterns (POPs) of length 4 and 5, discovering new results, consolidating existing literature, and proposing conjectures on pattern avoidance and Wilf-equivalence.

## Contribution

It provides 13 new enumerative results for classical patterns of length 4 and 5, consolidates sporadic literature results, and conjectures new connections to OEIS.

## Key findings

- 13 new enumerative results for patterns of length 4 and 5
- Collection of sporadic results in literature
- 6 conjectured connections to OEIS

## Abstract

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.