Identities of Eulerain and ordered Stirling numbers over a multiset

TL;DR

This paper generalizes identities involving Eulerian and ordered Stirling numbers over multisets, introduces q-analogs, and computes these numbers, expanding combinatorial understanding.

Contribution

It provides new generalized identities and q-analogs for Eulerian and ordered Stirling numbers over multisets, along with explicit computations.

Findings

Generalized identities for Eulerian and Stirling numbers over multisets

Introduction of q-analogs for these generalized identities

Explicit computation methods for these numbers over multisets

Abstract

By considering Eulerian numbers and ordered Stirling numbers of the second and third kinds over a multiset, we generalize identities of Eulerian numbers and Stirling numbers of the second and third kinds and provide $q$-analogs of these generalizations. Using these generalizations, we also compute Eulerian numbers and ordered Stirling numbers of the second kind over a multiset.