Scattered subsets
A set \(S \subseteq \{1, \dots, n\}\) is ``\(h\)-scattered'' if, for any \(s, t \in S\), \(|s - t|\geq h\). For each \(h\), \(n\), let \(f^{(h)}_n\) be the number of \(h\)-scattered subsets of \(\{1, \dots, n\}\). This article explores elementary properties of \(f^{(h)}_n\), and of the consequent generating functions, and of related coefficients and generating functions. Central to the article is the increasingly popular notion of a ``species'': abusing notation a little bit, for each \(h\) and each \(n\), let Sc\(^{(h)}(\{1, \dots, n\})\) be the set of \(h\)-scattered subsets of \(\{1, \dots, n\}\), and we can call the function Sc\(^{(h)}\) the ``linear species of the \(h\)-scattered subsets.'' Using generating functions, the article readily develops identities for these and related numbers, and for \(q\)-analogues of these numbers.
scientific article; zbMATH DE number 1934677