Touchard $C$-polynomials and polynomials quasi-orthogonal to them

We consider the Touchard $C$-polynomials closely related to the cycle indices of symmetric groups and introduce the so-called $M$-polynomials. We demonstrate that the $C$- and $M$-polynomials constitute a quasi-orthogonal system. For the partition polynomials under consideration, we give a series of recurrence relations.