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Motivated by Gauss's first proof of the Fundamental Theorem of Algebra, we study the topology of harmonic algebraic curves. By the maximum principle, a harmonic curve has no bounded components; its topology is determined by the combinatorial data of a noncrossing matching. Similarly, every complex polynomial gives rise to a related combinatorial object that… (More)

In this paper, we prove that the dimension of the space spanned by the characters of the symmetric powers of the standard n-dimensional representation of S n is asymptotic to n 2 /2. This is proved by using generating functions to obtain formulas for upper and lower bounds, both asymptotic to n 2 /2, for this dimension. In particular, for n ≥ 7, these… (More)

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