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A bijection between edges of the Turán graph and irreducible elements in the dominance order lattice

Nathanaël Hassler

TL;DR

This work establishes a direct correspondence between the edges of the $(n,p)$-Turán graph and the meet-irreducible elements of the dominance lattice on $[1,p]$-bounded compositions of $n$, yielding a new bridge between graph theory and lattice theory. The authors construct explicit reciprocal maps $\\Psi_n^p$ and $\\Phi_n^p$ to match $E(\\mathcal{T}_n^p)$ with $\\textsc{mi}_n^p$ (and similarly with join-irreducibles), and show $|\\textsc{mi}_n^p|=a_p(n)$, where $a_p(n)$ satisfies a Turán-type recurrence $a_p(n)=a_p(n-1)+\left\lfloor\left(1-\frac{1}{p}\right) n\right\rfloor$. Using this bijection, the paper computes asymptotic averages for three statistics on $\\textsc{mi}_n^p$: the number of parts, the first part, and the number of weak records, providing explicit leading-term behavior in $n$ and $p$. The results connect extremal graph theory with distributive lattice representations of bounded compositions, offering precise asymptotic insights into the structure of meet-irreducibles in this lattice.

Abstract

In this paper we build a bijection between the meet-irreducible elements of the lattice of the compositions of $n$ with parts in $[1,p]$ equipped with the dominance order, and the edges of the $(n,p)$-Turán graph. Using this bijection, we then compute asymptotically the average value of some statistics on those meet-irreducible compositions.

A bijection between edges of the Turán graph and irreducible elements in the dominance order lattice

TL;DR

This work establishes a direct correspondence between the edges of the -Turán graph and the meet-irreducible elements of the dominance lattice on -bounded compositions of , yielding a new bridge between graph theory and lattice theory. The authors construct explicit reciprocal maps and to match with (and similarly with join-irreducibles), and show , where satisfies a Turán-type recurrence . Using this bijection, the paper computes asymptotic averages for three statistics on : the number of parts, the first part, and the number of weak records, providing explicit leading-term behavior in and . The results connect extremal graph theory with distributive lattice representations of bounded compositions, offering precise asymptotic insights into the structure of meet-irreducibles in this lattice.

Abstract

In this paper we build a bijection between the meet-irreducible elements of the lattice of the compositions of with parts in equipped with the dominance order, and the edges of the -Turán graph. Using this bijection, we then compute asymptotically the average value of some statistics on those meet-irreducible compositions.
Paper Structure (6 sections, 7 theorems, 42 equations, 2 figures, 1 table)

This paper contains 6 sections, 7 theorems, 42 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The bijection
  3. Some statistics on $\textsc{mi}_n^p$
  4. The number of parts
  5. The first part
  6. The number of weak records

Key Result

Lemma 2.1

The number of upper covers of a composition $x\in\mathbb{F}_n^p$ is the number of consecutive patterns $ab$ in $x$ with $1\leq a\leq p-1$ and $2\leq b\leq p$, plus one if $x$ has the suffix $i1$ for $1\leq i\leq p-1$.

Figures (2)

  • Figure 1: The $(8,3)$-Turán graph $\mathcal{T}_8^3$.
  • Figure 2: The lattice $\mathbb{F}_5^3$ and the set $\textsc{mi}_5^3$ of its meet-irreducible elements.

Theorems & Definitions (15)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Remark 2.5
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 5 more