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The Redei-Berge function in noncommuting variables

Stefan Mitrovic

Abstract

Recently, Stanley and Grinberg introduced a symmetric function associated to digraphs, called the Redei-Berge symmetric function. This function, however, does not satisfy the deletion-contraction property, which is a very powerful tool for proving various identities using induction. In this paper, we introduce an analogue of this function in noncommuting variables which does have such property. Furthermore, it specializes to the ordinary Redei-Berge function when the variables are allowed to commute. This modification allows us to further generalize properties that are already proved for the original function and to deduce many new ones.

The Redei-Berge function in noncommuting variables

Abstract

Recently, Stanley and Grinberg introduced a symmetric function associated to digraphs, called the Redei-Berge symmetric function. This function, however, does not satisfy the deletion-contraction property, which is a very powerful tool for proving various identities using induction. In this paper, we introduce an analogue of this function in noncommuting variables which does have such property. Furthermore, it specializes to the ordinary Redei-Berge function when the variables are allowed to commute. This modification allows us to further generalize properties that are already proved for the original function and to deduce many new ones.
Paper Structure (3 sections, 19 theorems, 50 equations)

This paper contains 3 sections, 19 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Redei-Berge function in noncommuting variables

Key Result

Theorem 2.2

S If $X=(V, E)$ is a digraph, then with $\varphi(\sigma)=\sum_{\gamma}(\ell (\gamma)-1)$, where the summation runs over all cycles $\gamma$ of $\sigma$ that are cycles in $X$ and $\ell(\gamma)$ denotes the length of the cycle $\gamma$. Consequently, $U_X$ is a symmetric function.

Theorems & Definitions (31)

  • Definition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Definition 3.1
  • ...and 21 more