On lattices associated with maximal graphical partitions

The aim of this paper is to describe, for a given graphical partition $\lambda$ of weight $2m$ and rank $r$, the set of all maximal graphical partitions $\mu$ of weight $2m$ that dominate $\lambda$. To do this, it is enough to find the set of heads of such partitions. Theorem 1 states that, for any natural number $t$, the set of heads of all maximal graphical partitions $\mu$ of weight $2m$ and rank $t$ dominating $\lambda$ forms an interval of the integer partition lattice if such partitions $\mu$ of rank $t$ exist. Algorithms are also provided for finding the smallest and largest elements of this interval.