# Jay Pantone

## Two Examples of Unbalanced Wilf-Equivalence

with Alex Burstein

We prove that the set of patterns $$\{1324,3416725\}$$ is Wilf-equivalent to the pattern $$1234$$ and that the set of patterns $$\{2143,3142,246135\}$$ is Wilf-equivalent to the set of patterns $$\{2413,3142\}$$. These are the first known unbalanced Wilf-equivalences for classical patterns between finite sets of patterns.

##### Acknowledgments

The authors would like to thank Miklós Bóna for his close reading of the paper, helpful comments in clarifying the exposition, and pointing out the connection of this paper to [4].

##### Open Questions
• Conjecture 3.1: The classes $$\text{Av}(2413)$$ and $$\text{Av}(2143,246135)$$ are Wilf-equivalent.
• Conjecture: (due to Eric Egge) The class $$\text{Av}(2413,3142)$$ is Wilf-equivalent to the class $$\text{Av}(2143, 3142, \beta)$$ for any $$\beta \in \{246135,$$$$254613, 263514, 362415, 461325, 524361, 546132, 614352\}$$.
This was proved by Bloom and Burstein.