Combinatorics and Arithmetic for Physics: special days 2023

Collection Combinatorics and Arithmetic for Physics: special days 2023

Organisateur(s) Gérard H. E. DUCHAMP, Maxim KONTSEVICH, Gleb KOSHEVOY, Sergei NECHAEV and Karol A. PENSON
Date(s) 15/11/2023 - 17/11/2023
URL associée https://www-lipn.univ-paris13.fr/~duchamp/Conferences/CAP10_2023.html
00:00:00 / 00:00:00
18 27

The Redei–Berge symmetric function of a directed graph

De Darij Grinberg

In 1934, Laszlo Redei observed a peculiar property of tournaments (directed graphs that have an arc between every pair of distinct vertices): Each tournament has an odd number of Hamiltonian paths. In 1996, Chow introduced the “path-cycle symmetric function” of a directed graph, a symmetric function in two sets of arguments, which was later used in rook theory. We study Chow’s symmetric function in the case when the y-variables are 0. In this case, we give new nontrivial expansions of the function in terms of the power-sum basis; in particular, we find that it is p-positive as long as the directed graph has no 2-cycles. We use our expansions to reprove Redei’s theorem and refine it to a mod-4 congruence. Joint work with with Richard P. Stanley.

Informations sur la vidéo

  • Date de publication 22/11/2023
  • Institut IHES
  • Audience Chercheurs
  • Format MP4

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