Extraordinary Subsets

Abstract

For $n\ge 1$, we let $a_n$ count the number of nonempty subsets $S$ of $\{1,2,3,\dots ,n\}=[n]$, where the size of $S$ equals the minimal element of $S$. Such a subset is called an extraordinary subset of $[n]$, and we find that $a_n=F_n$, the $n$th Fibonacci number. Then, for $n\ge k\ge 1$, we let $a(n,k)$ count the number of times the integer $k$ appears among these $a_n$ extraordinary subsets of $n$. Here we have $a(n,k)=a(n-1,k)+a(n-2,k-1)$, for $n\ge 3$ and $n>k\ge 2$. Formulas and properties for $t_n=\sum _{k=1}^{n}a(n,k)$ and $s_n=\sum _{k=1}^{n}ka(n,k)$ are given for $n\ge 1$. Finally, for fixed $n\ge 1$, we find that the sequence $a(n,k)$ is unimodal and examine the maximum element for the sequence. In this context, the Catalan numbers make an entrance.