Motivation - I was thinking about calculating the integrals from An interesting integral expression for $\pi^n$? using old plain Riemann sums. There, one needs integrating over that part of $[0,1]^n$ where the coordinates are more than $\varepsilon$ apart from each other, so I thought maybe I can choose a grid for the multidimensional Riemann integral with these $\varepsilon$s somehow built in.
Then a natural thing to try is this: $$ \left\{(k_1,...,k_n)\in\mathbb Z^n\mid k_i\ne k_j\mod n\text{ for $i\ne j$}\right\} $$ (and then scale down to $\varepsilon$)
The above set is clearly a(n affine) lattice. Have you seen it before? What symmetries does it have? Is it listed in any nomenclature and how to find it?
Specifically I would like to have a more manageable enumeration of this lattice, - say, a nice basis. All bases I can think of are very unnatural, break symmetry terribly and result in messy formulas when I am trying to calculate anything with them.
