# Jay Pantone

## On the Growth of Merges and Staircases of Permutation Classes

with Michael Albert and Vincent Vatter

There is a well-known upper bound on the growth rate of the merge of two permutation classes. Curiously, there is no known merge for which this bound is not achieved. Using staircases of permutation classes, we provide sufficient conditions for this upper bound to be achieved. In particular, our results apply to all merges of principal permutation classes. We end by demonstrating how our techniques can be used to reprove a result of Bóna.

##### Open Questions
• Question 4: Is it the case that $$\text{gr}(\mathcal{C} \odot \mathcal{D}) = \left(\sqrt{\text{gr}(\mathcal{C})} + \sqrt{\text{gr}(\mathcal{D})}\right)^2$$ for every pair of classes $$\mathcal{C}$$ and $$\mathcal{D}$$ with proper growth rates?
• Question 10: Is $$\text{gr}(\text{Grid}(\text{Av}(21) \;\; \text{Av}(21)) \odot \text{Av}(21)) = 3 + 2\sqrt{2}$$ ?
• Section 2: The $$(\text{Av}(21), \text{Av}(21))$$-staircase is exactly the class $$\text{Av}(321)$$, counted by the Catalan numbers with growth rate 4. Peculiarly, the $$(\text{Av}(12), \text{Av}(12))$$-staircase appears to be infinitely based and have a growth rate around 4.5189296. What is the generating function of the $$(\text{Av}(12), \text{Av}(12))$$-staircase? (Is it algebraic?) What is its growth rate?