Two Self-Dual Lattices of Signed Integer Partitions

In this paper we study two self-dual lattices of signed integer partitions, $D(m,n)$ and $E(m,n)$, which can be considered also sub-lattices of the lattice $L(m,2n)$, where $L(m,n)$ is the lattice of all the usual integer partitions with at most $m$ parts and maximum part not exceeding $n$. We also introduce the concepts of $k$-covering poset for the signed partitions and we show that $D(m,n)$ is $1$-covering and $E(m,n)$ is $2$-covering. We study $D(m,n)$ and $E(m,n)$ as two discrete dynamical models with some evolution rules. In particular, the $1$-covering lattices are exactly the lattices definable with one outside addition rule and one outside deletion rule. The $2$-covering lattices have further need of another inside-switch rule.