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The Rearrangement Conjecture states that if two words over $\mathbb{P}$ areWilf-equivalent in the factor order on $\mathbb{P}^\ast$ then they arerearrangements of each other. We introduce the notion of strongWilf-equivalence and prove that if two words over $\mathbb{P}$ are stronglyWilf-equivalent then they are rearrangements of each other. We furtherconjecture that Wilf-equivalence implies strong Wilf-equivalence.

On the Rearrangement Conjecture for Generalized Factor Order Over $\mathbb{P}$