A New Approach to Order Polynomials of Labeled Posets and Their Generalizations

This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and hypergraphs, relational structures, and others. The second part advertises five topics in general enumeration: 1. counting lattice points in lattice polytopes, 2. growth of context-free languages, 3. holonomicity (i.e., P-recursiveness) of nu...

Partially ordered sets (posets) have a universal appearance as an abstract structure in many areas of mathematics. Though, even their explicit enumeration remains unknown in general, and only the counts of all partial orders on sets of up to 16 unlabelled elements have been calculated to date, see sequence A000112 in the OEIS. In this work, we study automorphisms of posets in order to formulate a classification by local symmetries. These symmetries give rise to a division o...

We introduce a family of quasisymmetric functions called {\em Eulerian quasisymmetric functions}, which have the property of specializing to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type....

In recent years we have worked on a project involving poset topology, various analogues of Eulerian polynomials, and a refinement of Richard Stanley's chromatic symmetric function. Here we discuss how Stanley's ideas and results have influenced and inspired our own work.

This paper examines operad structures derived from poset matrices by formulating a set of new construction rules for poset matrices. In this direction, eleven different partial composition operations will be introduced as the basis for the construction of poset matrices of any given size by extending the combinatorial setting of species of structures to poset matrices. Three of these partial composition operations are shown to define an operad structure for poset matrices. Th...

We give an in-depth study of the Tchebyshev transforms of the first and second kind of a poset, recently discovered by Hetyei. The Tchebyshev transform (of the first kind) preserves desirable combinatorial properties, including Eulerianess (due to Hetyei) and EL-shellability. It is also a linear transformation on flag vectors. When restricted to Eulerian posets, it corresponds to the Billera, Ehrenborg and Readdy omega map of oriented matroids. One consequence is that nonnega...

In this paper, we derive some interesting symmetric properties for the geenralized Euler numbers and polynomials.

We study the explicit formula of Euler numbers and polynomials of higher order

We consider the matrix ${\frak Z}_P=Z_P+Z_P^t$, where the entries of $Z_P$ are the values of the zeta function of the finite poset $P$. We give a combinatorial interpretation of the determinant of ${\frak Z}_P$ and establish a recursive formula for this determinant in the case in which $P$ is a boolean algebra.

The main purpose of this paper is to introduce and investigate a new class of generalized Bernoulli polynomials and Euler polynomials based on the q-integers. The q-analogues of well-known formulas are derived. The q-analogue of the Srivastava--Pint\'er addition theorem is obtained. We give new identities involving q-Bernstein polynomials.