Faà di Bruno’s formula, lattices, and partitions. (English) Zbl 1127.11023
Summary: The coefficients of \(g^{(s)}\) in expanding the \(r\)-th derivative of the composite function \(g \circ f\) by Faà di Bruno’s formula, is determined by a Diophantine linear system, which is proved to be equivalent to the problem of enumerating partitions of a finite set of integers attached to \(r\) and \(s\) canonically.
MSC:
| 11D45 | Counting solutions of Diophantine equations |
| 11H06 | Lattices and convex bodies (number-theoretic aspects) |
| 11D04 | Linear Diophantine equations |
| 05A17 | Combinatorial aspects of partitions of integers |
Keywords:
derivatives of composite functions; Diophantine equations with lower and upper bounds; integer partitions; knapsack systems; latticesReferences:
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