Journal of Integer Sequences, Vol. 22 (2019), Article 19.6.5 |

Department of Mathematics

University of Illinois at Urbana-Champaign

Champaign, IL 61820

USA

**Abstract:**

A finite subset of the natural numbers is *weak-Schreier* if min
*S* ≥ |*S*|,
*strong-Schreier* if min *S* > |*S*|,
and *maximal* if min *S* = |*S*|.
Let *M*_{n} be the number of weak-Schreier sets with
largest element *n* and let
(*F*_{n})_{n≥-1}
denote the Fibonacci sequence. A finite set is said to be
*Zeckendorf* if it does
not contain two consecutive natural numbers.
Let *E*_{n} be the number of
Zeckendorf subsets of {1,2, …, *n*}. It is well-known that
*E*_{n} = *F*_{n+2}.
In this paper, we first show four other ways to generate the
Fibonacci sequence from counting Schreier sets.
For example, let *C*_{n}
be the number of weak-Schreier subsets of {1,2,…,*n*}.
Then *C*_{n} =
*F*_{n+2}.
To understand why *C*_{n} = *E*_{n}, we provide a bijective mapping to
prove the equality directly. Next, we prove linear recurrence relations
among the number of Schreier-Zeckendorf sets. Lastly, we discover the
Fibonacci sequence by counting the number of subsets of
{1,2,…,*n*} such that two consecutive elements
in increasing order always differ by an
odd number.

(Concerned with sequence A000045.)

Received June 26 2019; revised version received August 31 2019;
September 2 2019.
Published in *Journal of Integer Sequences*,
September 3 2019.

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