Weighted Schreier-type Sets and the Fibonacci Sequence

Abstract

For a finite set $A\subset \mathbb{N}$ and $k\in \mathbb{N}$, let ${\omega }_k(A)={\sum }_{i\in A,i\ne k}1$. For each $n\in \mathbb{N}$, define $$a_{k, n}\ =\ |\{E\subset \mathbb{N}\,:\, E = \emptyset\mbox{ or } ω_k(E) < \min E\leqslant \max E\leqslant n\}|.$$ First, we prove that $$a_{k,k+\ell} \ =\ 2F_{k+\ell},\mbox{ for all }\ell\geqslant 0\mbox{ and }k\geqslant \ell+2,$$ where $F_n$ is the $n$th Fibonacci number. Second, we show that $$|\{E\subset \mathbb{N}\,:\, \max E = n+1, \min E > ω_{2,3}(E), \mbox{ and }|E|\neq 2\}|\ =\ F_{n},$$ where ${\omega }_{2,3}(E)={\sum }_{i\in E,i\ne 2,3}1$.

14 pages