A Second Type Of Higher Order Generalised Geometric Polynomials and Higher Order Generalised Euler Polynomials

A generalised Euler-Poincaré formula for associahedra

We derive a formula for the number of flip-equivalence classes of tilings of an $n$-gon by collections of tiles of shape dictated by an integer partition $λ$. The proof uses the Euler-Poincaré formula; and the formula itself generalises the Euler-Poincaré formula for associahedra.

A Combinatorial Analysis Of Higher Order Generalised Geometric Polynomials: A Generalisation Of Barred Preferential Arrangements

A barred preferential arrangement is a preferential arrangement, onto which in-between the blocks of the preferential arrangement a number of identical bars are inserted. We offer a generalisation of barred preferential arrangements by making use of the generalised Stirling numbers proposed by Hsu and Shiue (1998). We discuss how these generalised barred preferential arrangements offer a unified combinatorial interpretation of geometric polynomials. We also discuss asymptotic properties of these numbers.

加权Ehrhart级数与MacMahon公式的B型类似物

Weighted Ehrhart series and a type-$\mathsf{B}$ analogue of a formula of MacMahon

本文通过多面体的$h^*$-多项式给出了带符号多重集排列上主指标与下降统计量联合分布的生成多项式公式，并建立了A、B型广义欧拉数与多面体之间的联系，进而将结果推广到着色排列。

欧拉分支贪婪拟阵的贪婪拟阵多项式根无关性的几何证明

A geometric proof for the root-independence of the greedoid polynomial of Eulerian branching greedoids

该研究通过定义正则有向拟阵的根多胞体，证明欧拉分支贪婪拟阵的贪婪拟阵多项式等价于对应图拟阵对偶根多胞体的$h^*$-多项式，从而给出其根无关性的几何证明，还得到边反转不改变该多项式的结论。

Investigating Exponential and Geometric Polynomials with Euler-Seidel Algorithm

In this paper we use Euler-Seidel matrices method to find out some properties of exponential and geometric polynomials and numbers. Some known results are reproved and some new results are obtained.

Gamma-positivity of variations of Eulerian polynomials

An identity of Chung, Graham and Knuth involving binomial coefficients and Eulerian numbers motivates our study of a class of polynomials that we call binomial-Eulerian polynomials. These polynomials share several properties with the Eulerian polynomials. For one thing, they are $h$-polynomials of simplicial polytopes, which gives a geometric interpretation of the fact that they are palindromic and unimodal. A formula of Foata and Schützenberger shows that the Eulerian polynomials have a stronger property, namely $γ$-positivity, and a formula of Postnikov, Reiner and Williams does the same for…