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The Redei-Berge Hopf algebra of digraphs

Vladimir Grujić, Tanja Stojadinović

Abstract

In a series of recent talks Richard Stanley introduced a symmetric function associated to digraphs called the Redei-Berge symmetric function. This symmetric function enumerates descent sets of permutations corresponding to digraphs. We show that such constructed symmetric function arises from a suitable structure of combinatorial Hopf algebra on digraphs. The induced Redei-Berge polynomial satisfies the deletion-contraction property which makes it similar to the chromatic polynomial. The Berge's classical result on the number of Hamiltonian paths in digraphs is a consequence of the reciprocity formula for the Redei-Berge polynomial.

The Redei-Berge Hopf algebra of digraphs

Abstract

In a series of recent talks Richard Stanley introduced a symmetric function associated to digraphs called the Redei-Berge symmetric function. This symmetric function enumerates descent sets of permutations corresponding to digraphs. We show that such constructed symmetric function arises from a suitable structure of combinatorial Hopf algebra on digraphs. The induced Redei-Berge polynomial satisfies the deletion-contraction property which makes it similar to the chromatic polynomial. The Berge's classical result on the number of Hamiltonian paths in digraphs is a consequence of the reciprocity formula for the Redei-Berge polynomial.
Paper Structure (6 sections, 17 theorems, 65 equations)

This paper contains 6 sections, 17 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Basics from combinatorial Hopf algebras
  3. Hopf algebra of digraphs
  4. A generating function for X-descent sets
  5. The Redei-Berge polynomial
  6. Tournaments

Key Result

Lemma 3.3

The operations of taking the complementary and the opposite digraphs satisfy and consequently descend to commuting involutional anti-isomorphisms of the algebra $\mathcal{D}$.

Theorems & Definitions (36)

  • Example 3.1
  • Definition 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Definition 3.5
  • Proposition 3.6
  • proof
  • Definition 4.1
  • Proposition 4.2: GS
  • proof
  • ...and 26 more