A note on monotone subsequences and the RS image of invariant random permutations with macroscopic number of fixed points

The work of Vershik and Kerov [1977], Logan and Shepp [1977] established that the shape of the scaled random young diagram in Russian notation, as determined by the Plancherel measure, converges to a deterministic shape. In this article, we focus on the scenario where the number of fixed points is substantial. We provide evidence that, subject to specific requirements on the total number of cycles, the limiting shape is a scaled version of the Vershik-Kerov-Logan-Shepp limiting shape. Additionally, we identify certain limiting regimes that resemble those in Chapuy, Louf, and Walsh [2022]. Furthermore, we enhance the existing results on Tracy-Widom universality classes for $\beta \in \{ 1, 2, 4\}$ for monotone subsequences.