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)3k+1∑n=0∞{{n+kn}}xn analogous to that of the Eulerian numbers, where {{nk}}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 nn approaches infinity is the normal distribution.