Legendre–Stirling permutations

We first give a combinatorial interpretation of Everitt, Littlejohn, and Wellman’s Legendre–Stirling numbers of the first kind. We then give a combinatorial interpretation of the coefficients of the polynomial ( 1 − x ) 3 k + 1 ∑ n = 0 ∞ { { n + k n } } x n analogous to that of the Eulerian numbers, where { { n k } } are Everitt, Littlejohn, and Wellman’s Legendre–Stirling numbers of the second kind. Finally we use a result of Bender to show that the limiting distribution of these coefficients as n approaches infinity is the normal distribution.