New Modular Symmetric Function and its Applications: Modular s-Stirling Numbers

Abstract

In this paper, we consider generalizations of the Stirling number of the first and the second kind by using a specialization of a new family of symmetric functions. We give combinatorial interpretations for these symmetric functions by means of weighted lattice path and tilings. We also present some new convolutions involving the complete and elementary symmetric functions. Additionally, we introduce different families of set partitions to give combinatorial interpretations for the modular s-Stirling numbers.