Scaled Arndt compositions

Abstract: In 2013, Joerg Ardnt observed that integer compositions $c_1 + c_2 + \cdots = n$ with $c_{2i-1} > c_{2i}$ for each positive $i$ are counted by the Fibonacci numbers. This was confirmed by the speaker and Tangboonduangjit in 2022 and we explored generalizations of this pair-wise condition including $c_{2i-1} > c_{2i} + k$ for an affine parameter $k$. In the current work, a collaboration with Augustine Munagi, we consider scaling parameters, integers $s$ and $t$, and resolve some cases of the general condition $sc_{2i-1} > tc_{2i} + k$. Techniques include generating functions and combinatorial proofs.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)