Matrices over a finite field having fixed rank and restricted support are a natural qanalogue of rook placements on a board. We develop this q-rook theory by defining a corresponding analogue of the hit numbers. Using tools from coding theory, we show that these q-hit and q-rook numbers obey a variety of identities analogous to the classical case. We also explore connections to earlier q-analog...

We introduce a new type of graph drawing called “rook-drawing”. A rook-drawing of a graph G is obtained by placing the n nodes of G on the intersections of a regular grid, such that each row and column of the grid supports exactly one node. This paper focuses on rook-drawings of planar graphs. We first give a linear algorithm to compute a planar straight-line rook-drawing for outerplanar graphs...

A saw-toothed chessboard, or STC for short, is a kind of chessboard whose boundary forms two staircases from left down to right without any hole inside it. A rook at square (i, j) can dominate the squares in row i and in column j . The rook problem of an STC is to determine the minimum number of rooks that can dominate all squares of the STC. In this paper, we model an STC by two particular gra...

Rook polynomials are a powerful tool in the theory of restricted permutations. It is known that the rook polynomial of any board can be computed recursively, using a cell decomposition technique of Riordan. [13] In this paper, we give a new decomposition theorem, which yields a more efficient algorithm for computing the rook polynomial. We show that, in the worst case, this block decomposition ...

this paper has investigated mamluks egypt`s militarism influence on rook of hesam al-din & nasser qalawoon. the authors have concluded in this study that the generalissimo and military officers due to the monopoly on their extensive apanage areas and abundant revenues could limit the power of the sultan and even to the throne have greed sometimes; because much of the land, i.e 20 carats of 24 c...

Abstract. For an element w in the Weyl algebra generated by D and U with relation DU = UD + 1, the normally ordered form is w = ∑ ci,jU D . We demonstrate that the normal order coefficients ci,j of a word w are rook numbers on a Ferrers board. We use this interpretation to give a new proof of the rook factorization theorem, which we use to provide an explicit formula for the coefficients ci,j ....

The wealth of beautiful combinatorics that arise in the representation theory of the symmetric group is well-known. In this paper, we analyze the representations of a related algebraic structure called the rook monoid from a combinatorial angle. In particular, we give a combinatorial construction of the irreducible representations of the rook monoid. Since the rook monoid contains the symmetric...

1/i if i + j ≥ n + 1 0 otherwise, has eigenvalues 1,−1/2, 1/3, . . . , (−1)/n. In Section 2, we establish this result in two ways: (1) by explicitly exhibiting eigenvectors via an odd-looking combinatorial identity, and (2) by giving a matrix that conjugates Tn into an upper triangular matrix. In Section 3, we use the knowledge of the eigenvalues of Tn to study two related random walks, most co...