Journal of Combinatorics

Volume 1 (2010)

Number 1

A symmetrical Eulerian identity

Pages: 29 – 38



Fan Chung (University of California at San Diego)

Ron Graham (University of California at San Diego)

Don Knuth (Stanford University)


We give three proofs for the following symmetrical identity involvingbinomial coefficients $\binom{n}{m}$ and Eulerian numbers$\big\langle{n\atop m}\big\rangle$:\[\sum_{k} \binom {a+b}{k}\left< \begin{matrix}k\\a-1\end{matrix}\right> =\sum_{k} \binom {a+b} k\left< \begin{matrix} k \\b-1\end{matrix} \right>\]for any positive integers $a$ and $b$ (where we take$\big\langle{0\atop0}\big\rangle = 0$). We also show how this fitsinto a family of similar (but more complicated) identities for Euleriannumbers.

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