There is a well-known upper bound due to Claesson, Jelínek and Steingrímsson for the growth rate of the merge of two permutation classes. Curiously, there is no known merge for which this bound is not achieved. Using linear algebraic techniques and appealing to the theory of Toeplitz matrices, we provide sufficient conditions for the growth rate to equal this upper bound. In particular, our results apply to all merges of principal permutation classes. We end by demonstrating how our techniques relate to the results of Bóna.
Michael Albert. Jay Pantone. Vincent Vatter. "On the growth of merges and staircases of permutation classes." Rocky Mountain J. Math. 49 (2) 355 - 367, 2019. https://doi.org/10.1216/RMJ-2019-49-2-355