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

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.

💡 Research Summary

The paper establishes a concrete bijection between the meet‑irreducible elements of the dominance‑order lattice of integer compositions with parts bounded by p and the edges of the (n, p)‑Turán graph. After introducing the necessary notation—n mod p for the remainder, the Turán graph Tₚⁿ as the complete p‑partite graph on vertices {1,…,n} where two vertices are adjacent precisely when their residues modulo p differ—the authors recall that the number of edges is aₚ(n)=½(1−1/p)n²−(n mod p)(p−(n mod p))/2p.

The set Fₚ(n) consists of all compositions of n whose parts lie in