Study of $p^{k}$-Eulerian polynomials and $p^{k}$-Fibonacci numbers for every odd prime $p$ and $k\geq0$

TL;DR

This paper introduces $p^{k}$-Eulerian polynomials and $p^{k}$-Fibonacci numbers derived from paths in $p$-Bratteli diagrams, revealing their properties, generating functions, and recurrence relations for odd prime $p$ and $k \\geq 0.

Contribution

It defines and analyzes $p^{k}$-Eulerian polynomials and $p^{k}$-Fibonacci numbers, establishing their combinatorial and algebraic properties for the first time.

Findings

Coefficients of $p^{k}$-Eulerian polynomials count paths with specific descents.

Total descents of paths relate to $p^{k}$-Fibonacci numbers.

Derivative of the polynomial at 1 equals the $p^{k}$-Fibonacci number.

Abstract

In this paper, we define the notion of descent for the paths in the $p$-Bratteli diagram. This leads to the definition of $p^{k}$-Eulerian polynomials, whose coefficients count the number of paths with a given number of descents. We provide a method for constructing the $p^{k}$-Eulerian polynomials at each vertex. Furthermore, we compute the total number of descents of all paths ending at a given vertex as the corresponding $p^{k}$-Fibonacci numbers. We show that the derivative of the $p^{k}$-Eulerian polynomial evaluated at 1 for a fixed vertex equals the corresponding $p^{k}$-Fibonacci number. Finally, we discuss the generating function for the sequence of $p^{k}$-Fibonacci numbers and the recurrence relations they satisfy.