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- W4384261753 abstract "Let $D=left( V,Aright) $ be a digraph with $n$ vertices, where each arc $ain A$ is a pair $left( u,vright) $ of two vertices. We study the emph{Redei--Berge symmetric function} $U_{D}$, defined as the quasisymmetric function% [ sum L_{operatorname*{Des}left( w,Dright) , n}inoperatorname*{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,Dright) , n}:=sum_{substack{i_{1}leq i_{2}leqcdotsleq i_{n};i_{p}<i_{p+1}text{ for each }ptext{ 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." @default.
- W4384261753 created "2023-07-14" @default.
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- W4384261753 date "2023-07-10" @default.
- W4384261753 modified "2023-10-17" @default.
- W4384261753 title "The Redei--Berge symmetric function of a directed graph" @default.
- W4384261753 doi "https://doi.org/10.48550/arxiv.2307.05569" @default.
- W4384261753 hasPublicationYear "2023" @default.
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