A bijection between edges of the Tur\'an graph and irreducible elements in the dominance order lattice

TL;DR

This paper establishes a bijection linking the structure of compositions under dominance order to Turán graph edges, enabling asymptotic analysis of related statistics.

Contribution

It introduces a novel bijection between meet-irreducible compositions and Turán graph edges, facilitating new combinatorial insights.

Findings

Bijection between meet-irreducible compositions and Turán graph edges

Asymptotic computation of average statistics on compositions

Enhanced understanding of the lattice structure in combinatorics

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 $(n,p)$-Tur\'an graph. Using this bijection, we then compute asymptotically the average value of some statistics on those meet-irreducible compositions.