Pattern Avoidance in Task-Precedence Posets

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 represent warehouse package fulfillment by robots, in which case avoidance of both 231 and 321 ensures we never stack two heavier packages on top of a lighter package.