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

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