We characterise the permutations π such that the elements in the closed lower Bruhat interval [id, π] of the symmetric group correspond to nontaking rook configurations on a skew Ferrers board. It turns out that these are exactly the permutations π such that [id, π] corresponds to a flag manifold defined by inclusions, studied by Gasharov and Reiner. Our characterisation connects the Poincaré p...

The path of the rook is a curve γ (possibly, self-intersecting) inside an 8 × 8square. The length of γ should be at least 64 since all squares are visited. Next, I assumed that the rook turned each time by 90 degrees (and never 180). Then the total turn of γ is nπ/2, where n is the number of moves. Thus I traded the original problem for the following one: What is the smallest total turn of a cu...

The set of n by n upper-triangular nilpotent matrices with entries in a finite field Fq has Jordan canonical forms indexed by partitions λ ` n. We study a connection between these matrices and non-attacking q-rook placements, which leads to a combinatorial formula for the number Fλ(q) of matrices of fixed Jordan type as a weighted sum over rook placements. Résumé. L’ensemble des matrices triang...

Given k labelings of a finite d-dimensional cubical grid, define the combined distance between two labels to be the sum of the l1-distance between the two labels in each labeling. We want to construct k labelings which maximize the minimum combined distance between any two labels. When d = 1, this can be interpreted as placing n non-attacking rooks in a k-dimensional chessboard of size n in suc...