A Lucas analogue of Eulerian numbers

TL;DR

This paper introduces a recursive method to define Lucas analogues of classical combinatorial numbers, including Eulerian, Stirling, and Motzkin numbers, demonstrating their polynomial nature and palindromic properties.

Contribution

It provides the first recursive framework for Lucas analogues of key combinatorial numbers, expanding their algebraic and combinatorial understanding.

Findings

Lucas Eulerian numbers are polynomials with nonnegative coefficients.

Lucas analogues of Stirling and Motzkin numbers are also polynomial and palindromic.

Recursive formulas facilitate the computation of these Lucas analogues.

Abstract

The generalized Lucas numbers are polynomials in two variables with nonnegative integer coefficients. Lucas versions of some combinatorial numbers with known formulas in terms of quotient and products of nonnegative integers have been recently given by replacing the integers in those formulas with their corresponding Lucas analogues. We instead use a recursive approach. In this sense, we give a recursive formula for Lucas-Narayana numbers derived from a recent formula in terms of Lucasnomials (the explicit Lucas version of binomial numbers). We propose a recursive definition for a Lucas analogue of the classical Eulerian numbers, which shows immediately that they are polynomials in two variables with nonnegative integer coefficients. We prove that they are palindromic like their standard counterparts. The recursive approach allows us to give Lucas analogues of many relevant combinatorial constants. In particular, Lucas versions for both Stirling numbers of the second kind and Motzkin numbers are presented.