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

dmtcs:1515 - Discrete Mathematics & Theoretical Computer Science, June 24, 2016, Vol. 18 no. 2, Permutation Patterns 2015
Pattern Avoidance in Task-Precedence Posets

Authors: Paukner, Mitchell and Pepin, Lucy and Riehl, Manda and Wieser, Jarred

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.


Source : oai:arXiv.org:1511.00080
Volume: Vol. 18 no. 2, Permutation Patterns 2015
Section: Permutation Patterns
Published on: June 24, 2016
Submitted on: June 22, 2016
Keywords: Mathematics - Combinatorics


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