A symmetrical q-Eulerian identity

Abstract

We find a $q$-analog of the following symmetrical identity involving binomial coefficients $\left(\genfrac{}{}{0ex}{}{n}{m}\right)$ and Eulerian numbers $A_{n,m}$, due to Chung, Graham and Knuth [{\it J. Comb.}, {\bf 1} (2010), 29--38]: {equation*} \sum_{k\geq 0}\binom{a+b}{k}A_{k,a-1}=\sum_{k\geq 0}\binom{a+b}{k}A_{k,b-1}. {equation*} We give two proofs, using generating function and bijections, respectively.

Séminaire Lotharingien de Combinatoire, vol. 67(2012), Article B67c11 pages

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• Recurrence relations for binomial-Eulerian polynomials
• On some colored Eulerian quasisymmetric functions
• Around the $q$-binomial-Eulerian polynomials