A Class of Combinatorial Functions for Eulerian Numbers

Abstract

In this note, we introduce a new class of functions pertaining to binomial coefficients $\binom{n}{m}$ and Eulerian numbers $\left < \! \begin{array}{c} n \\ m \end{array} \!\right}$ which arise in the recent study of descents in permutations. Given positive integers $a$ and $b$, let $f_i(x) = 2^{−i} \binom{a+b}{i} \left <\! \begin{array}{c} i \\ x \end{array} \!\right}$ and $f(x) = \sum_i f_i(x)$. Based on the generating function methods, some identities involving $f_i$ and $f$ are provided. Keywords: Eulerian number; Binomial coefficient; Generating function. 2010 Mathematics Subject Classification: 05A19.