# Jay Pantone

## The Enumeration of Permutations Avoiding 3124 and 4312

We find the generating function for the class of all permutations that avoid the patterns $$3124$$ and $$4312$$ by showing that it is an inflation of the union of two geometric grid classes.

##### Acknowledgments

The author is very grateful to his advisor, Vince Vatter, for introducing him to this problem and for advice that significantly improved the presentation of this paper. Additionally, the author would like to thank Michael Albert for providing code that was useful in developing these arguments.

##### OEIS Sequences
• A000045 - $$\operatorname{Simples}\left(\mathcal{G}_3\right)$$
• A000079 - $$\operatorname{Simples}\left(\mathcal{G}_1\right)$$
• A083323 - $$\mathcal{G}_1$$
• A165534 - $$\operatorname{Av}\left(3124,4312\right)$$
• A226430 - $$\operatorname{Simples}\left(\operatorname{Av}\left(3124,4312\right)\right)$$
• A226431 - $$\operatorname{Simples}\left(\mathcal{G}_2\right)$$
• A226432 - $$\mathcal{G}_2$$
• A226433 - $$\mathcal{G}_3$$
• A226434 - Sum decomposable permutations in $$\operatorname{Av}\left(3124,4312\right)$$
• A228769 - Skew sum decomposable permutations in $$\operatorname{Av}\left(3124,4312\right)$$
• A228770 - Sum indecomposable permutations in $$\operatorname{Av}\left(3124,4312\right)$$
• A228771 - Skew sum indecomposable permutations in $$\operatorname{Av}\left(3124,4312\right)$$