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arXiv · 2506.16438

Eulerian-type polynomials over Stirling permutations and box sorting algorithm

Shi-Mei Ma·Jun-Ying Liu·Jean Yeh·Yeong-Nan Yeh
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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.

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Shi-Mei Ma, Jun-Ying Liu, Jean Yeh, Yeong-Nan Yeh. 2025-06-19. Eulerian-type polynomials over Stirling permutations and box sorting algorithm. https://arxiv.org/abs/2506.16438

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