Weighted Schreier-type Sets and the Fibonacci Sequence

TL;DR

This paper explores weighted Schreier-type sets and reveals their deep connection to Fibonacci numbers, providing explicit formulas and combinatorial results that link set properties to Fibonacci sequences.

Contribution

It establishes new Fibonacci-based formulas for weighted Schreier-type sets and analyzes their combinatorial structure, extending understanding of set enumeration related to Fibonacci numbers.

Findings

Proves that a_{k,k+ell} = 2F_{k+ell} for certain parameters.

Shows a Fibonacci count for specific subset properties involving max, min, and weights.

Connects set combinatorics with Fibonacci sequence properties.

Abstract

For a finite set $A\subset\mathbb{N}$ and $k\in \mathbb{N}$, let $\omega_k(A) = \sum_{i\in A, i\neq k}1$. For each $n\in \mathbb{N}$, define $$a_{k, n}\ =\ |\{E\subset \mathbb{N}\,:\, E = \emptyset\mbox{ or } \omega_k(E) < \min E\leqslant \max E\leqslant n\}|.$$ 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 $$|\{E\subset \mathbb{N}\,:\, \max E = n+1, \min E > \omega_{2,3}(E), \mbox{ and }|E|\neq 2\}|\ =\ F_{n},$$ where $\omega_{2,3}(E) = \sum_{i\in E, i\neq 2, 3}1$.