A Refinement of the Eulerian Numbers, and the Joint Distribution of $\pi (1)$ and Des($\pi$) in $S_n$

Given a permutation $\pi$ chosen uniformly from $S_n$, we explore the joint distribution of $\pi (1)$ and the number of descents in $\pi$. We obtain a formula for the number of permutations with $\text{des}(\pi )=d$ and $\pi (1)=k$, and use it to show that if $\text{des}(\pi )$ is fixed at $d$, then the expected value of $\pi (1)$ is $d+1$. We go on to derive generating functions for the joint distribution, show that it is unimodal if viewed correctly, and show that when $d$ is small the distribution of $\pi (1)$ among the permutations with $d$ descents is approximately geometric. Applications to Stein’s method and the Neggers-Stanley problem are presented.