On partition polynomials

We consider some partition polynomials such that partial Bell and Platonov polynomials, generalised $A$- and $B$-polynomials, Touchard $T$- and $C$-polynomials, and $P$-polynomials which compose a quasi-orthogonal system with $T$-polynomials. We investigate the properties of the coefficients of these polynomials, present recurrence relations and combinatorial interpretations. For $C$-polynomials, we give recurrence relations, generating functions, and explicit expressions.