The modular (r,s)-Stirling numbers of the first kind

We propose a new generalization of the [Formula: see text]-Stirling numbers of the first kind and their analogs. These numbers appear as specialization of a new class of symmetric function, and they can be seen as a natural generalizations of the [Formula: see text]-Stirling numbers of the first kind and their analogs. We also give a combinatorial interpretations of the classical numbers in terms of [Formula: see text]-tuples of permutations of [Formula: see text] with [Formula: see text] cycles where the first [Formula: see text] elements of each permutation are in distinct cycles, and using the inversion statistics on the cycles in the cases of the analogs numbers. Moreover, using the hyperharmonic numbers and the [Formula: see text]-Stirling numbers of the first kind new formulas and useful properties are established.