TITLE: Hexagonal rooks
QUESTION [22 upvotes]: Suppose you have a triangular chessboard of size $n$, whose "squares" are ordered triples $(x,y,z)$ of nonnegative integers that add up to $n$.  A rook can move to any other point that agrees with it in one coordinate -- for example, if you are on $(3,1,4)$ then you can move to $(2,2,4)$ or to $(6,1,1)$, but not to $(4,3,1)$.
What is the maximum number of mutually non-attacking rooks that can be placed on this chessboard?
More generally, is anything known about the graph whose vertices are these ordered triples and whose edges are rook moves?

REPLY [2 votes]: This has a connection with additive permutations, asked about here:
Are there enough additive permutations? .
http://oeis.org/A002047 has references to a similar problem involving a hexagonal board and counting the number of placements. (The Bennett and Potts reference calls the piece a 'brook'.) This number also counts the permutations.  A recent ArXiv paper http://arxiv.org/abs/1510.05987 appears to provide an asymptotic estimate.  Note that placement on a hexagonal board gives one for the triangle, but there may be placements in a triangle corner that are not realizable in the hexagon.
Gerhard "Cutting Corners For More Results" Paseman, 2017.03.13