A combinatorial interpretation of the generalized Fibonacci numbers
type: Item
instance of: scholarly article
title: A combinatorial interpretation of the generalized Fibonacci numbers
zbMATH Open document ID: 0884.11017
author: Emanuele Munarini
published in: Advances in Applied Mathematics
publication date: 1998-04-01T00:00:00Z
review text: The author considers the Fibonacci numbers of order \(k\), i.e., the numbers \(F^{(k)}_n\) defined by the recurrence relation \(F^{(k)}_{n+ k}= F_{n+k-1}^{(k)}+ F_{n+ k-2}^{(k)}+\cdots+ F_{n+1}^{(k)}+ F^{(k)}_n\) and, as in \textit{D. E. Knuth} [The art of computer programming, Vol. 3, Addison-Wesley (1974; Zbl 0302.68010)], by the initial conditions \(F^{(k)}_0=\cdots= F_{k-2}^{(k)}= 0\), \(F_{k-1}^{(k)}= 1\). He derives a combinatorial interpretation of them in the context of the linear species of \textit{A. Joyal} [Adv. Math. 42, 1-82 (1981; Zbl 0491.05007)] as the linear species of \(k\)-filtering partitions. He also proves a number of interesting identities.
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 1089803
zbMATH Keywords: Fibonacci numbers of higher order
zbMATH Keywords: filtering partitions
zbMATH Keywords: linear species
zbMATH Keywords: identities
reviewed by: Andreas N. Philippou
MaRDI profile type: Publication
full work available at URL: aama.1997.0531
OpenAlex ID: W1989224492
cites work: Q3826629
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DOI: 10.1006/AAMA.1997.0531
description: scientific article; zbMATH DE number 1089803