We are going to discuss recent progress on many problems in random matrix theory of a combinatorial nature, including several breakthroughs that solve long standing famous conjectures.

We present a systematic construction of probes into the dynamics isospectral ensembles Hamiltonians by notion Isospectral twirling, expanding scopes and methods ref. [1]. The relevant are those defined salient spectral probability distributions. Gaussian Unitary Ensembles (GUE) describes class quantum chaotic Hamiltonians, while spectra corresponding to Poisson Diagonal Ensemble (GDE) describe ...