A spline interpretation of Eulerian numbers

TL;DR

This paper establishes a novel connection between Eulerian numbers and B splines, providing explicit formulas and proving log-concavity of descent polynomial coefficients, offering a new perspective on their properties.

Contribution

It introduces a spline-based approach to analyze Eulerian numbers and descent polynomials, deriving explicit formulas and proving log-concavity.

Findings

Explicit formulas for refined Eulerian numbers using B splines

Proof of log-concavity of descent polynomial coefficients

A new approach to studying Eulerian numbers and descent polynomials

Abstract

In this paper, we explore the interrelationship between Eulerian numbers and B splines. Specifically, using B splines, we give the explicit formulas of the refined Eulerian numbers, and descents polynomials. Moreover, we prove that the coefficients of descent polynomials $D_d^n(t)$ are log-concave. This paper also provides a new approach to study Eulerian numbers and descent polynomials.