The Fibonacci partition triangles

TL;DR

This paper introduces Fibonacci partition triangles, which are similar to Pascal's triangle but based on additivity along hooks, revealing polynomial evaluations and relationships through differences along specific moves.

Contribution

It presents a novel Fibonacci partition triangle structure, connecting it to valued translation quivers and polynomial evaluations, expanding combinatorial and representation theory insights.

Findings

Fibonacci partition triangles are similar to Pascal's triangle but based on hook additivity.

Numbers in the triangles are given by evaluating polynomials.

The two triangles are related through differences along arrows and knight's moves.

Abstract

In two previous papers we have presented partition formulae for the Fibonacci numbers motivated by the appearance of the Fibonacci numbers in the representation theory of the 3-Kronecker quiver and its universal cover, the 3-regular tree. Here we show that the basic information can be rearranged in two triangles. They are quite similar to the Pascal triangle of the binomial coefficients, but in contrast to the additivity rule for the Pascal triangle, we now deal with additivity along hooks, or, equivalently, with additive functions for valued translation quivers. As for the Pascal triangle, we see that the numbers in these Fibonacci partition triangles are given by evaluating polynomials. We show that the two triangles can be obtained from each other by looking at differences of numbers, it is sufficient to take differences along arrows and knight's moves.