Ordinal and disjoint sums of partially ordered patterns

TL;DR

This paper extends the theory of partially ordered patterns by introducing ordinal and disjoint sums, leading to new Wilf-equivalence results and a classification of small patterns, confirming a recent conjecture.

Contribution

It develops the notions of ordinal and disjoint sums of labeled posets, generalizing previous work and establishing new Wilf-equivalence results for POPs.

Findings

Proves shape-Wilf-equivalence of specific pattern sets

Classifies small POPs with chain components

Confirms a conjecture from BCC30

Abstract

Partially ordered patterns (POPs) generalize the classical notion of permutation patterns within the framework of pattern avoidance. Building on recent work by Burstein, Han, Kitaev, and Zhang, which introduced the concept of shape-Wilf-equivalence of sets of patterns, we develop the notions of \emph{ordinal} and \emph{disjoint sums} of labeled posets. This framework enables us to reinterpret their main result as an ordinal sum analogue of the classical theorem by Backelin, West, and Xin. We establish analogous results for disjoint sums of POPs and further extend their results to prove Wilf-equivalence for classes of POPs that include isolated vertices. In particular, we prove the shape-Wilf-equivalence of the sets of patterns $\{123, 213, 312\}$ and $\{132, 231, 321\}$. Our proof strategy involves a bijection that filters through an encoding scheme for the transversals avoiding these patterns. We use these results to completely classify the partially ordered patterns of size $3,4,5$ whose connected components are all chains. This classification also confirms a conjecture posed by Dimitrov at the problem session of the British Combinatorics Conference 2024 (BCC30).