I have a problem $\mathcal{P}(s_1,\dots,s_5)$, $s_i\in\mathbb{N}$, which is invariant by
- circular permutations ($\mathcal{P}(s_1,s_2,s_3,s_4,s_5)=\mathcal{P}(s_2,s_3,s_4,s_5,s_1)=\dots$)
- symmetry: $\mathcal{P}(s_1,s_2,s_3,s_4,s_5)=\mathcal{P}(s_5,s_4,s_3,s_2,s_1)$
This means that to investigate $\mathcal{P}$, it suffices to focus on one tenth of $\mathbb{N}^5$. My goal is to describe such a tenth.
I can start by requiring $s_1$ to be the (or a) largest: $\forall i\in\{2,3,4,5\},\ s_1\geq s_i$. This is not enough: for example, $(5,1,5,2,0)$ and $(5,2,0,5,1)$ should not be both considered. I chose the following canonical form, which I can describe with words: sort all five circular permutations and their symmetries by decreasing order; select the first one. For example, the canonical form of $(1,5,1,2,5)$ is $(5,2,1,5,1)$; $(2,5,5,1,3)$ becomes $(5,5,2,3,1)$, etc.
How can I describe such part of the space where the canonical points lie? Can it be done in terms of inequalities?
Note: the group I am considering is isomorphic to the dihedral group $D_5$, but I don't see how this can help.