Combinatorics of Partial Derivatives

Abstract

The natural forms of the Leibniz rule for the $k$th derivative of a product and of Fa\`a di Bruno's formula for the $k$th derivative of a composition involve the differential operator $\partial^k/\partial x_1 ... \partial x_k$ rather than $d^k/dx^k$, with no assumptions about whether the variables $x_1,...,x_k$ 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 Fa\`a 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.