Combinatorics of Partial Derivatives

2006/01/07 by Michael Hardy · 2 citations
Mathematics · Physics and Astronomy · Chemistry · #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications #Advanced Mathematical Identities #Mathematics #Partial derivative #Combinatorics #Generalization #Enumeration #Cumulant #Operator (biology) #Derivative (finance) #Differential operator #Product (mathematics) #Partially ordered set #Pure mathematics #Mathematical analysis #Chemistry #Statistics

paper · pdf · doi:10.37236/1027

openalex publication_date 2006/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/15

Abstract

The natural forms of the Leibniz rule for the kth derivative of a product and of Faà di Bruno's formula for the kth derivative of a composition involve the differential operator ∂^k/∂ x₁ ⋯ ∂ x_k rather than d^k/dx^k, with no assumptions about whether the variables x₁,…,x_k are all distinct, or all identical, or partitioned into several distinguishable classes of indistinguishable variables. Coefficients appearing in forms of these identities in which some variables are indistinguishable are just multiplicities of indistinguishable terms (in particular, if all variables are distinct then all coefficients are 1). The computation of the multiplicities in this generalization of Faà di Bruno's formula is a combinatorial enumeration problem that, although completely elementary, seems to have been neglected. We apply the results to cumulants of probability distributions.

Citations

• A Multivariate Faa di Bruno Formula with Applications
• The Derivation of the Pattern Formulae of Two-Way Partitions from those of Simpler Patterns
• Prehistory of Faà di Bruno's Formula

Cited by

• On simplified application of multidimensional Savitzky-Golay filters and differentiators
• Faà di Bruno's formula [wikipedia]