Pattern Avoidance in Task-Precedence Posets
Mitchell Paukner, Lucy Pepin, Manda Riehl, Jarred Wieser

TL;DR
This paper extends pattern avoidance to task-precedence posets with three-level diamonds, using computational and combinatorial methods to enumerate avoiding permutations and explore applications in robotics and logistics.
Contribution
It introduces a new structure for pattern avoidance in posets, establishes bijections with Dyck paths, and derives generating functions for avoiding permutations.
Findings
Bijection between 132-avoiding diamonds and generalized Dyck paths
Derived generating functions for pattern-avoiding permutations
Applied results to robotics and warehouse package stacking
Abstract
We have extended classical pattern avoidance to a new structure: multiple task-precedence posets whose Hasse diagrams have three levels, which we will call diamonds. The vertices of each diamond are assigned labels which are compatible with the poset. A corresponding permutation is formed by reading these labels by increasing levels, and then from left to right. We used Sage to form enumerative conjectures for the associated permutations avoiding collections of patterns of length three, which we then proved. We have discovered a bijection between diamonds avoiding 132 and certain generalized Dyck paths. We have also found the generating function for descents, and therefore the number of avoiders, in these permutations for the majority of collections of patterns of length three. An interesting application of this work (and the motivating example) can be found when task-precedence posets…
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