Scattered subsets
type: Item
instance of: scholarly article
title: Scattered subsets
zbMATH Open document ID: 1017.05012
DOI: 10.1016/S0012-365X(02)00616-7
author: Emanuele Munarini
author: Norma Zagaglia Salvi
published in: Discrete Mathematics
publication date: 2003-06-25T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 1934677
zbMATH Keywords: linear species
zbMATH Keywords: cyclic species
zbMATH Keywords: scattered subsets of linear orders
zbMATH Keywords: scattered subsets of cycles
zbMATH Keywords: generalized Fibonacci numbers
zbMATH Keywords: generalized Lucas numbers
zbMATH Keywords: Cassini-like identities
zbMATH Keywords: generalizations of the ménage problem
reviewed by: Gregory Loren McColm
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1934677