I'm looking for background material on the majorization (aka dominance) order over $\mathbb{Z}^n$ (rather than over partitions).
Let $v=(\psi_0,\dots,\psi_{n-1})$ and $u=(\phi_0,\dots,\phi_{n-1})$ be two elements in $\mathbb{Z}^n$. Then $u$ majorizes/dominates $v$ if and only if $$\sum_{j=0}^i \psi_j\leq \sum_{j=0}^i \phi_j$$ for every $0\leq i \leq n-1$.
I'm interested in the covering relation of the poset, the rank function (it is a graded poset), distance function (corresponding to the covering relation/distance in the Hasse diagram) etc.
For some reason I couldn't find any reference specific to properties of this order over $\mathbb{Z}^n$, I hope I'm just missing something simple (maybe it's called differently over $\mathbb{Z}^n$ or something...).
Thanks
Answering my own question, based on Olivier's comments:
$\mathbb{Z}^n$ with the dominance order is isomorphic to $\mathbb{Z}^n$ with the point-wise $\leq$ order ($u\leq v$ iff $u_i\leq v_i$ for all $i$) through the isomorphism: $$\varphi : (\phi_0,\dots,\phi_{n-1}) \mapsto (\phi_0,\phi_0+\phi_1,\dots,\phi_0+\dots+\phi_{n-1})$$ With: $$\varphi^{-1} : (\phi_0,\dots,\phi_{n-1}) \mapsto (\phi_0,\phi_1-\phi_0,\phi_2-\phi_1,\dots,\phi_{n-1}-\phi_{n-2})$$
Covering Relation:
It's quite easy to see via $\varphi^{-1}$ that $\phi$ dominates $\psi$ iff $\phi_i = \psi_i +1$ and $\phi_{i+1} = \psi_{i+1} - 1$ for some $0\leq i\leq n-2$ or $\phi_{n-1} = \psi_{n-1} +1$ (and the rest are equal).
Rank: $$rank(\psi_0,\dots,\psi_{n-1})=\sum_{i=0}^{n-1}\sum_{j=0}^{i}\psi_j$$
Distance:
Using modularity of the posets, the distance between two elements in the Hasse diagram is the difference between the ranks of their join and their meet that can be easily calculated.