Derivatives are essentially integer partitions
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.
scientific article; zbMATH DE number 1506831