New modular symmetric function and its applications: Modular $s$-Stirling numbers

TL;DR

This paper introduces a new family of symmetric functions generalizing Stirling numbers, providing combinatorial interpretations and exploring their applications through convolutions and set partitions.

Contribution

It presents a novel modular symmetric function and defines modular s-Stirling numbers with combinatorial interpretations and new convolution identities.

Findings

Defined a new family of symmetric functions

Provided combinatorial interpretations for modular s-Stirling numbers

Derived new convolution formulas involving symmetric functions

Abstract

In this paper, we consider a generalization of the Stirling number sequence of both kinds by using a specialization of a new family of symmetric functions. We give combinatorial interpretations for this 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.