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Why do partitions occur in Faa di Bruno's chain rule for higher derivatives?
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It is well-known that the coefficients in Faa di Bruno's chain rule for higher derivatives can be expressed via numeration of partitions. It turns out that this has a natural form as a formula for the vector case. To this formula two proofs are presented, both "explaining" its form involving partitions: one as a purely algebraic fact, and one "from first principles" for the case of Frechet derivatives of mappings between Banach spaces.
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Cited by 2 Pith papers
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Discrete Fa\`a di Bruno via M\"obius Inversion
Composing discrete Taylor expansions and applying Möbius inversion yields a covering-indexed Faà di Bruno formula that deforms flatly into the classical partition form for Fréchet derivatives.
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Fa\`a di Bruno is Taylor Composition
The reduced Taylor polynomial of a composition equals the truncation of the composition of the reduced Taylor polynomials for C^k maps between Banach spaces.
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