Stirling permutation codes

The development of the theory of the second-order Eulerian polynomials began with the works of Buckholtz and Carlitz in their studies of an asymptotic expansion. Gessel-Stanley introduced Stirling permutations and provided combinatorial interpretations for the second-order Eulerian polynomials in terms of Stirling permutations. The Stirling permutations have been extensively studied by many researchers. The motivation of this paper is to develop a general method for finding equidistributed statistics on Stirling permutations. Firstly, we show that the up-down-pair statistic is equidistributed with the ascent-plateau statistic, and that the exterior up-down-pair statistic is equidistributed with the left ascent-plateau statistic. Secondly, we introduce the Stirling permutation code (called SP-code). A large number of equidistribution results follow from simple applications of the SP-codes. In particular, we find that six bivariable set-valued statistics are equidistributed on the set of Stirling permutations, and we generalize a classical result on trivariate version of the second-order Eulerian polynomial, which was independently established by Dumont and Bóna. Thirdly, we explore the bijections among Stirling permutation codes, perfect matchings and trapezoidal words. We then show the e-positivity of the enumerators of Stirling permutations by left ascent-plateaux, exterior up-down-pairs and right plateau-descents. In the final part, the e-positivity of the multivariate k-th order Eulerian polynomials is established, which improves a classical result of Janson-Kuba-Panholzer and generalizes a recent result of Chen-Fu. These e-positive expansions are derived from the combinatorial theory of context-free grammars.