Dually weighted Stirling-type sequences

TL;DR

This paper generalizes Stirling numbers using symmetric functions with two weights, unifying various known sequences and deriving new properties, relations, and combinatorial interpretations.

Contribution

It introduces a broad generalization of Stirling numbers involving symmetric functions and two weights, connecting multiple known sequences and deriving their fundamental properties.

Findings

Unified various Stirling-type sequences with a symmetric function framework

Derived recurrence relations, generating functions, and orthogonality relations

Provided combinatorial interpretations for special cases

Abstract

We introduce a generalization of the Stirling numbers via symmetric functions involving two weight functions. The resulting extension unifies previously known Stirling-type sequences with known symmetric function forms, as well as other sequences such as the $p,q$-binomial coefficients. Recurrence relations, generating functions, orthogonality relations, convolution formulas, and determinants of certain matrices involving the obtained sequences are derived. We also give combinatorial interpretations of certain cases in terms of colored partitions and permutations.