Combinatorics of Partial Derivatives

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 xₖ rather than dᵏ/dxᵏ, with no assumptions about whether the variables x₁, , xₖ 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.