Stirling Permutations, Legendre-Stirling Permutations, and Jacobi-Stirling Permutations

It's not difficult to show that for each nonnegative integer k, the Stirling number of the second kind S(n+k,n) is a polynomial of degree 2k in n. Building on this, some standard generatingfunctionology tells us that for each k, the generating function for S(n+k,n) is a rational function whose denominator is a power of 1-x. In 1978 Gessel and Stanley proved that the numerator of this generating function is itself the generating function for a particular family of permutations with respect to the number of descents. While we are probably most familiar with the fact that the Stirling numbers of the second kind count set partitions of [n] into k blocks, they also arise when we repeatedly apply a certain differential operator to a generic function. When we replace this differential operator with differential operators related to certain orthogonal polynomials, we obtain analogues of the Stirling numbers, called the Legendre-Stirling numbers and the Jacobi-Stirling numbers. In this talk I will describe these analogues, paying particular attention to analogues of Gessel and Stanley's work on the Stirling numbers.