Eulerian-type polynomials over Stirling permutations and box sorting algorithm

Abstract: It is well known that ascents, descents and plateaux are equidistributed over the set of classical Stirling permutations. Their common enumerative polynomials are the second-order Eulerian polynomials, which have been extensively studied by many researchers. This paper is divided into three parts. The first parts gives a convolution formula for the second-order Eulerian polynomials, which simplifies a result of Gessel. As an application, a determinantal expression for the second-order Eulerian polynomials is obtained. We then investigate the convolution formula of the trivariate second-order Eulerian polynomials. Among other things, by introducing three new statistics: proper ascent-plateau, improper ascent-plateau and trace, we discover that a six-variable Eulerian-type polynomial over a class of restricted Stirling permutations equals a six-variable Eulerian-type polynomial over signed permutations. By special parametrizations, we make use of Stirling permutations to give a unified interpretations of the $(p,q)$-Eulerian polynomials and derangement polynomials of types $A$ and $B$. The third part presents a box sorting algorithm which leads to a bijection between the terms in the expansion of $(cD)^nc$ and ordered weak set partitions, where $c$ is a smooth function in the indeterminate $x$ and $D$ is the derivative with respect to $x$. Using a map from ordered weak set partitions to standard Young tableaux, we find an expansion of $(cD)^nc$ in terms of standard Young tableaux. Combining this with grammars, we provide three interpretations of the second-order Eulerian polynomials.