What are Binary Permutohedrons Called? I saw permutohedrons, but this case is under a permutation without repetitions. Is there a special object where vertices are permutations of a binary vector, and edges still connect vertices where the last element has been swapped with another element in the vector? What properties would this object have, given that there are no guarantees that the object is still a closed shape?
 A: The polytope we get is called a hypersimplex and denoted $\Delta_{n,k}$.
But first, some clarifications on how we get it.
In the permutohedron, the edges are not chosen arbitrarily (and are not restricted to just swapping the last element with something). We actually treat all permutations of $(1,2,\dots,n)$ as points in $n$-dimensional space, and then take their convex hull. The edges then come out of looking at which points are geometrically adjacent.
If we took all $\binom nk$ permutations of a binary vector with $k$ $1$'s and $n-k$ $0$'s, then these would also be points in $n$-dimensional space. (As with the permutohedron, the polytope we get is $(n-1)$-dimensional, since it lies on the hyperplane $x_1 + x_2 + \dots + x_n = k$.) The correct edges to draw are between points which differ by swapping any $0$ with any $1$.
Here are some special cases:

*

*When $n=3$, we get a point for $k=0$ or $k=3$, and a triangle for $k=1$ and $k=2$.

*When $n=4$, we get a point for $k=0$ or $k=4$, a tetrahedron for $k=1$ and $k=3$, and an octahedron for $k=2$. The vertices of the octahedron are $$\{(0,0,1,1), (0,1,0,1), (0,1,1,0), (1,0,0,1), (1,0,1,0),(1,1,0,0\}.$$ Opposite vertices of the octahedron correspond to complementary pairs like $(0,1,0,1)$ and $(1,0,1,0)$.

*When $n=5$, we get a point for $k=0$ or $k=5$, a $5$-cell for $k=1$ or $k=4$, and a rectified $5$-cell for $k=2$ or $k=3$.

