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{\bf Michael Hardy }
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{\bf Combinatorics of Partial Derivatives}
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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 $\partial^k/\partial x_1 \cdots \partial x_k$
rather than $d^k/dx^k$, with no assumptions about
whether the variables $x_1,\dots,x_k$ 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.
\bye