Fibonacci colored compositions and applications

TL;DR

This paper explores Fibonacci colored compositions, establishing explicit bijections with various combinatorial objects and representing them as connected tilings, revealing new structural insights.

Contribution

It introduces a novel combinatorial framework linking Fibonacci colored compositions to multiple well-known structures through explicit bijections.

Findings

Established bijections with ternary and quaternary words

Connected compositions to spanning trees in ladder graphs

Provided tiling interpretations of Fibonacci colored compositions

Abstract

We study compositions whose parts are colored by subsequences of the Fibonacci numbers. We give explicit bijections between Fibonacci colored compositions and several combinatorial objects, including certain restricted ternary and quaternary words, spanning trees in the ladder graph, unimodal sequences covering an initial interval, and ordered-consecutive partition sequences. Our approach relies on the basic idea of representing the colored compositions as tilings of an $n$-board whose tiles are connected, and sometimes decorated, according to a suitable combinatorial interpretation of the given coloring sequence.