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

In this note we prove combinatorially some new formulas connecting poly-Bernoulli numbers with negative indices to Eulerian numbers.

We introduce the theory of normal ordered grammars, which gives a natural generalization of the normal ordering problem. To illustrate the main idea, we explore normal ordered grammars associated with the Eulerian polynomials and the second-order Eulerian polynomials. In particular, we present a normal ordered grammatical interpretation for the (cdes,cyc) (p,q)-Eulerian polynomials, where cdes and cyc are the cycle descent and cycle statistics, respectively. The exponential g...

In this paper we introduce the generalization of Multi Poly-Euler polynomials and we investigate some relationship involving Multi Poly-Euler polynomials. Obtaining a closed formula for generalization of Multi Poly-Euler numbers therefore seems to be a natural and important problem.

We consider the generating polynomial of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. B. Drake proved that this polynomial factors completely ...

Let $G=(V,E)$ be a simple graph with $V=\{1,2,\cdots,n\}$ and $\chi(G,x)$ be its chromatic polynomial. For an ordering $\pi=(v_1,v_2,\cdots,v_n)$ of elements of $V$, let $\delta_G(\pi)$ be the number of $i$'s, where $1\le i\le n-1$, with either $v_i<v_{i+1}$ or $v_iv_{i+1}\in E$. Let ${\cal W}(G)$ be the set of subsets $\{a,b,c\}$ of $V$, where $a<b<c$, which induces a subgraph with $ac$ as its only edge. We show that ${\cal W}(G)=\emptyset$ if and only if $(-1)^n\chi(G,-x)=\...

The $(P, \omega)$-partition generating function of a labeled poset $(P, \omega)$ is a quasisymmetric function enumerating certain order-preserving maps from $P$ to $\mathbb{Z}^+$. We study the expansion of this generating function in the recently introduced type 1 quasisymmetric power sum basis $\{\psi_\alpha\}$. Using this expansion, we show that connected, naturally labeled posets have irreducible $P$-partition generating functions. We also show that series-parallel posets ...

We introduce a series of numbers which serve as a generalization of Bernoulli, Euler numbers and binomial coefficients. Their properties are applied to solve a probability problem and suggest a statistical test for independence and identical distribution of random variables.

This paper surveys a comprehensive, although not exhaustive, sampling of graph polynomials with the goal of providing a brief overview of a variety of techniques defining a graph polynomial and then for decoding the combinatorial information it contains. The polynomials we discuss here are not generally specializations of the Tutte polynomial, but they are each in some way related to the Tutte polynomial, and often to one another. We emphasize these interrelations and explore...

P\'olya's enumeration theorem is concerned with counting labeled sets up to symmetry. Given a finite group acting on a finite set of labeled elements it states that the number of labeled sets up to symmetry is given by a polynomial in the number of labels. We give a new perspective on this theorem by generalizing it to partially ordered sets and order preserving maps. Further we prove a reciprocity statement in terms of strictly order preserving maps generalizing a classical ...

Cobweb posets uniquely represented by directed acyclic graphs are such a generalization of the Fibonacci tree that allows joint combinatorial interpretation for all of them under admissibility condition. This interpretation was derived in the source papers ([6,7] and references therein to the first author).[7,6,8] include natural enquires to be reported on here. The purpose of this presentation is to report on the progress in solving computational problems which are quite eas...