Combinatorial proof of Girard-Waring formula
Abstract
\noindent The well-known and celebrated identity of Girard (1629) and Waring (1762) states that
\[x^n+y^n=\sum_{m=0}^{\lfloor n/2\rfloor}(-1)^m\frac{n}{n-m}\binom{n-m}{m}\left(xy\right)^m\left(x+y\right)^{n-2m}\]
and can be easily proven algebraically (see H.W. Gould, The Girard--Waring power sum formulas for symmetric functions and Fibonacci sequences,} Fibonacci Quart. 37 (1999), no. 2, 135--140). In this note, we provide a combinatorial proof of this identity.
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