Contents Online
Journal of Combinatorics
Volume 1 (2010)
Number 1
A symmetrical Eulerian identity
Pages: 29 – 38
DOI: https://dx.doi.org/10.4310/JOC.2010.v1.n1.a2
Authors
Abstract
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.
Published 1 January 2010