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

M. Parvathi, A. Tamilselvi, D. Hepsi

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.

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

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.
Paper Structure (11 sections, 15 theorems, 54 equations, 4 figures)

This paper contains 11 sections, 15 theorems, 54 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation and definitions
  3. The p-Bratteli diagram
  4. Vertex set of the p-Bratteli diagram
  5. Edge Set of the p-Bratteli Diagram
  6. Inversions and descents of paths
  7. Inversions and sign balances of paths
  8. Descents of paths
  9. The p(k)-Fibonacci numbers
  10. The generating functions
  11. Acknowledgments

Key Result

Theorem 13

Let $k\geq0$, $r\geq 1$ with $s=r-k.$ The sign balance for every vertex of shape $\lambda(2r;(x_{(2s,k)},x_{(s,k)}+l))\in V^{2r}_{k}$ is zero.

Figures (4)

  • Figure 5:
  • Figure 6:
  • Figure 7:
  • Figure 8:

Theorems & Definitions (44)

  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 6
  • Remark 7
  • Remark 8
  • Definition 9
  • Definition 10
  • Definition 11
  • Definition 12
  • ...and 34 more