Derivatives are essentially integer partitions
type: Item
instance of: scholarly article
title: Derivatives are essentially integer partitions
zbMATH Open document ID: 0964.05009
DOI: 10.1016/S0012-365X(99)00412-4
author: Winston C. Yang
published in: Discrete Mathematics
publication date: 2001-07-16T00:00:00Z
review text: The author presents a new proof of Faà di Bruno's formula expressing the \(n\)th derivative of the composition of 2 functions in terms of the derivatives of each function. Instead of the usual approach of exponential generating functions, he introduces a matrix notation that simplifies derivatives and relates them to integer partitions. This enables him to generalize the formula to a composition of \(m>2\) functions. The author claims that calculating derivatives using matrix notation has much less overhead and is much faster, compared with a computer algebra system. In addition, he establishes the following curious identity: Let \(M(m,n)\) be the number of terms of the \(n\)th derivative of a composition of \(m\) functions. Then \(M(m+1, n)=\sum^n_{N=0} p(n,N)M(m,N)\), where \(p(n,k)\) denotes the number of integer partitions of \(n\) into \(k\) parts.
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 05A10
zbMATH DE Number: 1506831
zbMATH Keywords: Faà di Bruno's formula
zbMATH Keywords: derivatives
zbMATH Keywords: generating functions
zbMATH Keywords: integer partitions
zbMATH Keywords: matrix notation
zbMATH Keywords: computer algebra system
reviewed by: Ljuben Mutafchiev
MaRDI profile type: Publication
full work available at URL: s0012-365x(99)00412-4
OpenAlex ID: W2077510990
description: scientific article; zbMATH DE number 1506831