Combinatorics of Partial Derivatives

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 /∂x 1 · · · ∂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 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.

The combinatorics of weighted vector compositions

“…We now show that Theorem 4.1 (or equivalently Theorem 4.2) generalizes the main formula derived in [22]. Recall that a set partition of [n] = {1, .…”

“…Interpreting the last quantity as a sum over set partitions, using Lemma 4.4, with the m i as subsets of [N ] yields the formula (5) in [22]. Correspondingly, our representation in Theorem 4.2 is the direct analogue of the representation in [22] based on 'multiset partitions' (Corollary to Propositions 1 and 2 in [22] combined with Proposition 4 therein).…”

“…The formula of Faà di Bruno (1825-1888) describes the higher-order derivatives of a composite function G • F as a combinatorial sum of the derivatives of the individual functions G and F . Hardy [22] generalizes this formula to partial derivatives, arguing that treating variables in the derivatives as distinct is more natural. We provide an alternative derivation of the partial derivative formula which is based on interpreting G • F as the generating function for weighted vector compositions.…”

Interpolation properties of generalized plane waves

“…The numbering of the rows in matrices M C n and M n is set up such that the derivatives of smaller order appear higher in the matrix, which proves (11). Indeed (12) shows that any coefficient of M n is the sum of the corresponding coefficient in M C n plus a linear combination -which coefficients do not depend on the column that is considered but only on β and its derivatives evaluated at G -of terms that appear higher in the corresponding column of M n .…”

Computation of mixed Lie derivatives in nonlinear control

“…The kth derivative of a composition of two real functions can be expressed explicitly by Faà di Bruno's formula [4,6], which has been extended to the vector-valued case [7,8]. For a function h : M → R, the kth derivative h (k) (p) at p ∈ M is a covariant k-tensor and therefore a multilinear map.…”