Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2506.16438

Eulerian-type polynomials over Stirling permutations and box sorting algorithm

Shi-Mei Ma·Jun-Ying Liu·Jean Yeh·Yeong-Nan Yeh
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Shi-Mei Ma, Jun-Ying Liu, Jean Yeh, Yeong-Nan Yeh. 2025-06-26. Eulerian-type polynomials over Stirling permutations and box sorting algorithm. https://arxiv.org/abs/2506.16438

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Matching Complexes of Outerplanar Graphs

An outerplanar graph is a planar graph that has a planar drawing with all vertices on the unbounded face. The matching complex of a graph is the simplicial complex whose faces are subsets of disjoint edges of the graph. In this paper we prove that the matching complexes of outerplanar graphs are contractible or homotopy equivalent to a wedge of spheres. This extends known results about trees and polygonal line tilings.

math.CO↗
arXiv2026

Awesome graph parameters

For a graph $G$, we denote by $α(G)$ the size of a maximum independent set and by $ω(G)$ the size of a maximum clique in $G$. Our paper lies on the edge of two lines of research, related to $α$ and $ω$, respectively. One of them studies $α$-variants of graph parameters, such as $α$-treewidth or $α$-degeneracy. The second line deals with graph classes where some parameters are bounded by a function of $ω(G)$. A famous example of this type is the family of $χ$-bounded classes, where the chromatic number $χ(G)$ is bounded by a function of $ω(G)$. A Ramsey-type argument implies that if the $α$-variant of a graph parameter $ρ$ is bounded by a constant in a hereditary class $\mathcal{G}$, then $ρ$ is bounded by a function of $ω$ in $\mathcal{G}$. If the reverse implication also holds, we say that $ρ$ is awesome. Otherwise, we say that $ρ$ is awful. In the present paper, we identify a number of awesome and awful graph parameters, derive some algorithmic applications of awesomeness, and propose a number of open problems related to these notions.

math.CO↗
arXiv2026

Perfect matchings and $A_α$-spectral radius in 1-binding graphs

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For $α\in[0,1)$, we use $A_α(G)$ and $ρ_α(G)$ to denote the $A_α$-matrix and the $A_α$-spectral radius of $G$, respectively. The binding number $\mbox{bind}(G)$ of $G$ is defined by $\mbox{bind}(G)=\min\left\{\frac{|N_G(X)|}{|X|}:\emptyset\neq X\subseteq V(G),N_G(X)\neq V(G)\right\}$. If $\mbox{bind}(G)\geq1$, then $G$ is called 1-binding. A perfect matching in $G$ is a set of nonadjacent edges covering every vertex of $G$. Tutte proved that a graph $G$ of even order has a perfect matching if and only if $o(G-S)\leq|S|$ holds for every $S\subseteq V(G)$ [W. Tutte, The factorization of linear graphs, J. Lond. Math. Soc. 22 (1947) 107--111]. In this paper, we use Tutte's result to prove that a connected 1-binding graph $G$ of even order $n$ with $n\geq n(α)$ has a perfect matching unless $G=K_1\vee(K_{n-5}\cup K_3\cup K_1)$ if $ρ_α(G)\geqρ_α(K_1\vee(K_{n-5}\cup K_3\cup K_1))$, where $n(α)$ is defined as follows: $n(α)=\max\{18,\frac{2+8α}{1-2α}\}$ if $α\in[0,\frac{1}{2})$, and $n(α)=18$ if $α=\frac{1}{2}$.

math.CO↗