Extraordinary Subsets

For \(n \geq 1\), we let \(a_n\) count the number of nonempty subsets \(S\) of \(\{1,2,3,\ldots,n\} = [n]\), 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 \(n \geq k \geq 1\), 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 \(n \geq 3\) and \(n > k \geq 2\). 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 \(n \geq 1\). Finally, for fixed \(n \geq 1\), 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.