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An identity relating $n$-nacci numbers, partitions, and products of binomial coefficients

Dušan Dragutinović

TL;DR

The paper studies final types $ u$ and their associated partitions $ abla_ u$ to uncover a rich combinatorial bridge between partition structures and $n$-nacci sequences. It derives a binomial-product identity that expresses the $n$-nacci numbers as sums over partitions, generalizing Lucas's classic Fibonacci-binomial identity. A precise counting framework via $N( abla)$ connects final types to partitions, yielding a characterization of when a partition corresponds to a unique final type and establishing $F_n(g)=F_n(g-1)+\

Abstract

We study the combinatorial properties of final types, which are certain non-decreasing sequences of integers, together with the partitions naturally associated with them. As a consequence, we obtain an identity expressing the $n$-nacci numbers as sums of products of binomial coefficients over these partitions, generalizing the classical identity for $n = 2$ that expresses Fibonacci numbers in this way. We also examine how the partial order on the set of all partitions of a fixed integer induced by the ordering of final types compares with two natural partial orders on the same set.

An identity relating $n$-nacci numbers, partitions, and products of binomial coefficients

TL;DR

The paper studies final types and their associated partitions to uncover a rich combinatorial bridge between partition structures and -nacci sequences. It derives a binomial-product identity that expresses the -nacci numbers as sums over partitions, generalizing Lucas's classic Fibonacci-binomial identity. A precise counting framework via connects final types to partitions, yielding a characterization of when a partition corresponds to a unique final type and establishing $F_n(g)=F_n(g-1)+\

Abstract

We study the combinatorial properties of final types, which are certain non-decreasing sequences of integers, together with the partitions naturally associated with them. As a consequence, we obtain an identity expressing the -nacci numbers as sums of products of binomial coefficients over these partitions, generalizing the classical identity for that expresses Fibonacci numbers in this way. We also examine how the partial order on the set of all partitions of a fixed integer induced by the ordering of final types compares with two natural partial orders on the same set.
Paper Structure (12 sections, 16 theorems, 79 equations, 2 figures)

This paper contains 12 sections, 16 theorems, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Final types and partitions
  3. Final types
  4. Partitions defined by final types
  5. Relations between final types and partitions
  6. Motivation for studying these relationships
  7. Final types and $n$-nacci numbers
  8. On $n$-nacci numbers
  9. Relations between $F_n(g)$ and $n$-nacci numbers
  10. Comparing three partial orders of partitions
  11. Comparing $\leq_{ft}$, $\leq_{pp}$, and $\leq_{do}$
  12. A summary of the comparison of three partial orders

Key Result

Theorem 1

For any $g \geq 1$ and $n \geq 1$, the following identity holds:

Figures (2)

  • Figure 1: Partitions $\delta \vdash g$, for $g \in \{3, 4, 5\}$; $\delta \rightarrow \delta'$ if ${\delta} \leq_{pp} {\delta'}$.
  • Figure 2: Partitions $\delta$ of $g = 6$; $\delta \rightarrow \delta'$ if ${\delta} \leq_{ft} {\delta'}$.

Theorems & Definitions (34)

  • Theorem
  • Corollary
  • Proposition
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 24 more