Some combinatorial formulas for the partial r-Bell polynomials

The partial r-Bell polynomials generalize the classical partial Bell polynomials (coinciding with them when r = 0) by assigning a possibly different set of weights to the blocks containing the r smallest elements of a partition no two of which are allowed to belong to the same block. In this paper, we study the partial r-Bell polynomials from a combinatorial standpoint and derive several new formulas. We prove some general identities valid for arbitrary values of the parameters as well as establish formulas for some specific evaluations. Several of our results extend known formulas for the partial Bell polynomials and reduce to them when r = 0. Our arguments are largely combinatorial, and therefore provide, alternatively, bijective proofs of these formulas, many of which were shown by algebraic methods.