The Redei--Berge symmetric function of a directed graph

TL;DR

This paper introduces the Redei--Berge symmetric function for directed graphs, providing new formulas and properties, including positivity and integrality results, and connects these to classical theorems on Hamiltonian paths.

Contribution

It develops new formulas expressing the Redei--Berge symmetric function in terms of power-sum symmetric functions and establishes positivity and integrality properties, extending classical results.

Findings

U_D is p-integral and p-positive for digraphs without 2-cycles.

For tournaments, U_D can be expressed as a polynomial in specific p_{odd} with nonnegative coefficients.

Specializations recover Redei and Berge's theorems on Hamiltonian paths and a modulo-4 refinement.

Abstract

Let $D=\left( V,A\right) $ be a digraph with $n$ vertices, where each arc $a\in A$ is a pair $\left( u,v\right) $ of two vertices. We study the \emph{Redei--Berge symmetric function} $U_{D}$, defined as the quasisymmetric function% \[ \sum L_{\operatorname*{Des}\left( w,D\right) ,\ n}\in\operatorname*{QSym}. \] Here, the sum ranges over all lists $w=\left( w_{1},w_{2},\ldots ,w_{n}\right) $ that contain each vertex of $D$ exactly once, and the corresponding addend is% \[ L_{\operatorname*{Des}\left( w,D\right) ,\ n}:=\sum_{\substack{i_{1}\leq i_{2}\leq\cdots\leq i_{n};\\i_{p}<i_{p+1}\text{ for each }p\text{ satisfying }\left( w_{p},w_{p+1}\right) \in A}}x_{i_{1}}x_{i_{2}}\cdots x_{i_{n}}% \] (an instance of Gessel's fundamental quasisymmetric functions). While $U_{D}$ is a specialization of Chow's path-cycle symmetric function, which has been studied before, we prove some new formulas that express $U_{D}$ in terms of the power-sum symmetric functions. We show that $U_{D}$ is always $p$-integral, and furthermore is $p$-positive whenever $D$ has no $2$-cycles. When $D$ is a tournament, $U_{D}$ can be written as a polynomial in $p_{1},2p_{3},2p_{5},2p_{7},\ldots$ with nonnegative integer coefficients. By specializing these results, we obtain the famous theorems of Redei and Berge on the number of Hamiltonian paths in digraphs and tournaments, as well as a modulo-$4$ refinement of Redei's theorem.