Given positive integers $m,n$, let $A_{m,n}$ be an abstract simplicial complex where the vertices are the cells in an $m$-by-$n$ grid, and the faces are sets of cells with at most one cell in each row. I am searching for information on properties of this complex, so first I would like to know: does it have a name?
Note that the complex containing only the sets of cells with at most one cell in each row and at most one cell in each column is known as a chessboard complex. The facets of the chessboard complex are simplices with $\min(m,n)$ vertices (so they are $\min(m,n)-1$-dimensional). In contrast, the facets of my complex are simplices with $m$ vertices (so they are $m-1$-dimensional). What term describes my complex?
I asked Frederic Meunier, and he explained to me that the complex I described is simply the join $D_n^{*(2n-1)}$, where:
The join is constructed by taking $2n-1$ disjoint copies of $D_n$, and forming all unions of a single simplex from each copy. In each copy, each simplex is either $\emptyset$ or contains a single point. Therefore, if we treat each copy as a row and each point as a cell, we see that each union contains at most one cell from each row.
Note that the chessboard complex is the deleted join ${D_n}^{*(2n-1)}_{\Delta}$. A deleted join contains alll unions of disjoint simplices from each copy. Therefore, if we treat each copy as a row, and all copies of each element as a column, we see that each union contains at most one cell from each row and at most one cell from each column.