p^(k)-Fibonacci Numbers of the p-Bratteli Diagram for Every Odd Prime p and Integer k>=0

TL;DR

This paper explores the combinatorial structure of p-Bratteli diagrams for odd primes, introducing p^(k)-Fibonacci numbers via descents, and establishes recurrence relations, extending known sequences and discovering new Fibonacci-type sequences.

Contribution

It introduces p^(k)-Fibonacci numbers for p-Bratteli diagrams, generalizing existing sequences and providing new recurrence relations for these Fibonacci-type sequences.

Findings

Sign balance from inversions vanishes at all vertices.

Recurrence relations for p^(k)-Fibonacci numbers are derived.

For k=0, sequence matches OEIS A391520; for k>=1, new sequences are found.

Abstract

We study paths in the p-Bratteli diagram associated with hook partitions, where p is an odd prime. By comparing blocks along a path, we define inversions and descents. We prove that the sign balance derived from inversions vanishes at every vertex of the diagram. Using descents, we introduce the p^(k)-Fibonacci numbers and derive recurrence relations for them. For k=0, we recover the OEIS sequence A391520, while for k>=1 we obtain new families of Fibonacci-type sequences.