  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.
