Combinatorial Interpretation of General Eulerian Numbers

TL;DR

This paper extends the combinatorial interpretation of Eulerian numbers to a broader class defined on arithmetic progressions, enriching the understanding of their combinatorial structure.

Contribution

It provides the first combinatorial interpretation of general Eulerian numbers on arbitrary arithmetic progressions, expanding classical results.

Findings

General Eulerian numbers are interpreted combinatorially on arithmetic progressions

The work generalizes classical Eulerian number interpretations

New combinatorial models are introduced for these generalized numbers

Abstract

Since 1950s, mathematicians have successfully interpreted the traditional Eulerian numbers and $q-$Eulerian numbers combinatorially. In this paper, the authors give a combinatorial interpretation to the general Eulerian numbers defined on general arithmetic progressions { a, a+d, a+2d,...}.