Pattern avoidance in forests of binary shrubs

• Published in: Discrete mathematics and theoretical computer science Vol. 18 no. 2, Permutation...; no. Permutation Patterns
• Main Authors: Bevan, David, Levin, Derek, Nugent, Peter, Pantone, Jay, Pudwell, Lara, Riehl, Manda, Tlachac, ML
• Format: Journal Article
• Language: English
• Published: Discrete Mathematics & Theoretical Computer Science 21.07.2016
• Subjects: mathematics - combinatorics
• ISSN: 1365-8050; 1365-8050
• DOI: 10.46298/dmtcs.1322

We investigate pattern avoidance in permutations satisfying some additional restrictions. These are naturally considered in terms of avoiding patterns in linear extensions of certain forest-like partially ordered sets, which we call binary shrub forests. In this context, we enumerate forests avoiding patterns of length three. In four of the five non-equivalent cases, we present explicit enumerations by exhibiting bijections with certain lattice paths bounded above by the line $y=\ell x$, for some $\ell\in\mathbb{Q}^+$, one of these being the celebrated Duchon's club paths with $\ell=2/3$. In the remaining case, we use the machinery of analytic combinatorics to determine the minimal polynomial of its generating function, and deduce its growth rate.