  We first revisit the multi-allelic mutation-fitness balance problem,
especially when mutations obey a house of cards condition, where the
discrete-time deterministic evolutionary dynamics of the allelic frequencies
derives from a Shahshahani potential. We then consider multi-allelic
Wright-Fisher stochastic models whose deviation to neutrality is from the
Shahsha-hani mutation/selection potential. We next focus on the weak selection,
weak mutation cases and, making use of a Gamma calculus, we compute the
normalizing partition functions of the invariant probability densities
appearing in their Wright-Fisher diffusive approximations. Using these results,
Generalized Ewens sampling formulae (ESF) from the equilibrium distributions
are derived. We start treating the ESF in the mixed mutation/selection
potential case and then we restrict ourselves to the ESF in the simpler
house-of-cards mutations only situation. We also address some issues concerning
sampling problems from infinitely-many alleles weak limits.
