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