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 \(Des(\pi) = d\) and \(\pi(1) = k\), and use it to show that if \(Des(\pi)\) 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.
1970-2025 CP (Manitoba, Canada) unless otherwise stated.