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.