Statistical Distributions and $q$-Analogues of $k$-Fibonacci Numbers

TL;DR

This paper introduces $q$-analogues of $k$-Fibonacci numbers derived from weighted tilings, connecting them to permutation and set partition statistics, and extends classical identities to these new $q$-analogues.

Contribution

It develops general $q$-analogues of $k$-Fibonacci identities based on tile weights, linking combinatorial structures with permutation and set partition statistics.

Findings

Derived $q$-analogues of $k$-Fibonacci identities.

Connected $q$-analogues to permutation and set partition statistics.

Provided specific identities for particular statistical weights.

Abstract

We study $q$-analogues of $k$-Fibonacci numbers that arise from weighted tilings of an $n\times1$ board with tiles of length at most $k$. The weights on our tilings arise naturally out of distributions of permutations statistics and set partitions statistics. We use these $q$-analogues to produce $q$-analogues of identities involving $k$-Fibonacci numbers. This is a natural extension of results of the first author and Sagan on set partitions and the first author and Mathisen on permutations. In this paper we give general $q$-analogues of $k$-Fibonacci identities for arbitrary weights that depend only on lengths and locations of tiles. We then determine weights for specific permutation or set partition statistics and use these specific weights and the general identities to produce specific identities.