September 21: Mark Conger, Refined Eulerian Numbers
The Eulerian Number $A_{n,d}$ is the number of permutations of $n$ letters with $d$ descents. The Eulerian Numbers have been studied for a long time, and there is a nice recurrence relation to compute them, a formula in terms of binomial coefficients, generating functions, and a proof that they are well approximated by the normal distribution. This talk will derive similar results for the ``refined'' Eulerian Numbers $A_{n,d,k}$, the number of permutations $\pi$ which have $d$ descents and $\pi(1)=k$.