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Generalized tableaux over arbitrary digraphs and their associated differential equations

Luis Martínez, Antonio Vera López, Antonio Vera Pérez, Beatriz Vera Pérez, Olga Basova

Abstract

We revisit the concepts of acyclic orderings and number of acyclic orderings of acyclic digraphs in terms of dispositions and counters for arbitrary multidigraphs. We prove that when we add a sequence of nested directed paths to a directed graph there is a unique polynomial such that the generatrix function of the family of counters is the product of the polynomial and the exponential function. We give an application, by considering a kind of digraphs arranged in rows introduced by the authors in a previous paper, called dispositional digraphs, in the particular case in which the digraph has two rows, to obtain new families of linear differential equations of small order whose coefficients are polynomials of small degree which admit polynomial solutions. In particular, we obtain a new differential equation associated to Catalan numbers, and the corresponding associated polynomials, which are solution of this differential equation; we term them Catalan differencial equation and Catalan polynomials, respectively. We prove that the Catalan polynomials obtained when we connect the directed path to the second vertex of the lower row of the digraph are orthogonal polynomials for an appropriate weight function. We characterize the digraphs that maximize the counter of connected dispositional digraphs and we find a new differential equation associated to these digraphs. We introduce also dispositions and counters in any multidigraph with non-strict inequalities in the dispositions, and we find new differential equations associated to some of them.

Generalized tableaux over arbitrary digraphs and their associated differential equations

Abstract

We revisit the concepts of acyclic orderings and number of acyclic orderings of acyclic digraphs in terms of dispositions and counters for arbitrary multidigraphs. We prove that when we add a sequence of nested directed paths to a directed graph there is a unique polynomial such that the generatrix function of the family of counters is the product of the polynomial and the exponential function. We give an application, by considering a kind of digraphs arranged in rows introduced by the authors in a previous paper, called dispositional digraphs, in the particular case in which the digraph has two rows, to obtain new families of linear differential equations of small order whose coefficients are polynomials of small degree which admit polynomial solutions. In particular, we obtain a new differential equation associated to Catalan numbers, and the corresponding associated polynomials, which are solution of this differential equation; we term them Catalan differencial equation and Catalan polynomials, respectively. We prove that the Catalan polynomials obtained when we connect the directed path to the second vertex of the lower row of the digraph are orthogonal polynomials for an appropriate weight function. We characterize the digraphs that maximize the counter of connected dispositional digraphs and we find a new differential equation associated to these digraphs. We introduce also dispositions and counters in any multidigraph with non-strict inequalities in the dispositions, and we find new differential equations associated to some of them.
Paper Structure (7 sections, 65 theorems, 224 equations, 16 figures)

This paper contains 7 sections, 65 theorems, 224 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Dispositions and counters
  3. Companion polynomial of a digraph
  4. Some differential equations associated to companion polynomials of digraphs
  5. Differential equation associated to a dispositional digraph with two lines
  6. Staircase differential equations
  7. Non-strict dispositions in digraphs

Key Result

Proposition 1

Let $\frak G=(V,A)$ be a digraph. The relation on $V$ defined by is an equivalence relation.

Figures (16)

  • Figure 1: Staircase digraph of order $5$
  • Figure 2: In-degree 1. the oval in each part represents the dispositional digraph induced by $V(\frak G)-\{v\}$
  • Figure 3: In-degree 2. the ovals in each part represent the connected components of $V(\frak G)-\{v\}$
  • Figure 4: Star digraph of order $5$
  • Figure 5: Digraph $\frak G$
  • ...and 11 more figures

Theorems & Definitions (84)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Proposition 1
  • Definition 9
  • ...and 74 more