Pattern avoidance in ordered set partitions and words
Anisse Kasraoui

TL;DR
This paper explores the enumeration of ordered set partitions avoiding permutation patterns, establishing identities with pattern-avoiding words, determining growth rates, and analyzing specific pattern cases.
Contribution
It introduces an explicit identity linking pattern-avoiding partitions to words, enabling transfer of results and detailed analysis of pattern avoidance.
Findings
Derived asymptotic growth rates for pattern-avoiding partitions.
Confirmed a conjecture on sequence variation for pattern-avoiding partitions.
Analyzed the number of partitions avoiding length-3 patterns.
Abstract
We consider the enumeration of ordered set partitions avoiding a permutation pattern, as introduced by Godbole, Goyt, Herdan and Pudwell. Let be the number of ordered set partitions of into blocks that avoid a permutation pattern . We establish an explicit identity between the number and the numbers of words avoiding the inverse of . This identity allows us to easily translate results on pattern-avoiding words obtained in earlier works into equivalent results on pattern-avoiding ordered set partitions. In particular, \emph{(a)} we determine the asymptotic growth rate of the sequence for every positive and every permutation pattern , \emph{(b)} we partially confirm a conjecture of Godbole et al. concerning the variation of the sequences , \emph{(c)} we undertake a…
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