The development of the theory of second-order Euler polynomials began with the work of Buckholtz and Carlitz in the study of asymptotic expansions. Gessel-Stanley introduced the Stirling permutation and presented a combinatorial interpretation of the second-order Euler polynomials. Recently, there has been an increasing interest in the properties of Stirling permutations. The motivation for this paper is to develop a general method for finding equidistribution statistics for Stirling permutations. First, we show that the up-down pair statistic is equidistributed with the ascending plateau statistic and the outer up-down pair statistic is equivariance with the left ascending plateau statistic. Next, we introduce the Stirling permutation code. A simple application yields several equally distributed results. In particular, we find that the six bivariate set-valued statistics are equally distributed over the set of Stirling permutations. As an application, it extends the classic results independently established by Dumont and Bona. Third, we examine Stirling permutation codes, exact matches, and bijections between trapezoidal words. We then show the e-positivity of the enumerator for Stirling permutations with left ascending plateaus, external up-down pairs and right plateau descending. In the final part, the e-positivity of multivariate $k$ order Euler polynomials is established, improving the Janson-Kuba-Panholzer result and generalizing the recent result of Chen-Fu.
