Let k ≥ 2, m ≥ 5 and n = mk be integers. By finding bounds for certain rook polynomials, we identify the k×n Latin rectangles with the most extensions to (k+1)×n Latin rectangles. Equivalently, we find the (n− k)-regular subgraphs of Kn,n which have the greatest number of perfect matchings, and the (0, 1)-matrices with exactly k zeroes in every row and column which maximise the permanent. Witho...

How many perfect matchings are contained in a given bipartite graph? An exercise in Godsil’s 1993 Algebraic Combinatorics solicits proof that this question’s answer is an integral involving a certain rook polynomial. Though not widely known, this result appears implicitly in Riordan’s 1958 An Introduction to Combinatorial Analysis. It was stated more explicitly and proved independently by S.A. ...

We present a number of conjectures involving rook polynomials having only real zeros. Many of these generalize a previous conjecture of the author, K. Ono, and D. G. Wagner, namely that if A is a real n n matrix which is weakly increasing down columns, then the permanent of zA + Jn has only real zeros. In some cases these conjectures are motivated by factorization theorems for Ferrers boards. S...

Utilizing elliptic weights, we construct an elliptic analogue of rook numbers for Ferrers boards. Our elliptic rook numbers generalize Garsia and Remmel’s q-rook numbers by two additional independent parameters a and b, and a nome p. The elliptic rook numbers are shown to satisfy an elliptic extension of a factorization theorem which in the classical case was established by Goldman, Joichi and ...

In this article, we solve a problem of enumerative combinatorics addressed and left open in [16]. The initial question is formulated in terms of paths on an infinite three-dimensional chessboard. The 3D chessboard being identified with N, a 3D rook is a piece which is allowed to move parallelly to one of the three axes. The general objective is to count paths (i.e., finite sequences of moves) o...

The Orientale basin is the youngest and best-preserved multiring impact basin on the Moon, having experienced only modest modification by subsequent impacts and volcanism. Orientale is often treated as the type example of a multiring basin, with three prominent rings outside of the inner depression: the Inner Rook Montes, the Outer Rook Montes, and the Cordillera. Here we use gravity data from ...