Rook Polynomials Research Articles

Development of Non-Stretchable Triples with Numerable Features

Abstract: In this paper, some non-extendable triples involving Rook Polynomials, Hermite Polynomials and Laguerre Polynomials with appropriate properties are accomplished. Besides, it is showed that these triples cannot be prolonged into quadruples with the help of property of congruence.

Partitions of matrix spaces with an application to [formula omitted]-rook polynomials

We study the row-space partition and the pivot partition on the matrix space Fqn×m. We show that both these partitions are reflexive and that the row-space partition is self-dual. Moreover, using various combinatorial methods, we explicitly compute the Krawtchouk coefficients associated with these partitions. This establishes MacWilliams-type identities for the row-space and pivot enumerators of linear rank-metric codes. We then generalize the Singleton-like bound for rank-metric codes, and introduce two new concepts of code extremality. Both of them generalize the notion of MRD code and are preserved by trace-duality. Moreover, codes that are extremal according to either notion satisfy strong rigidity properties analogous to those of MRD codes. As an application of our results to combinatorics, we give closed formulas for the q-rook polynomials associated with Ferrers diagram boards. Moreover, we exploit connections between matrices over finite fields and rook placements to prove that the number of matrices of rank r over Fq supported on a Ferrers diagram is a polynomial in q, whose degree is strictly increasing in r. Finally, we investigate the natural analogues of the MacWilliams Extension Theorem for the rank, the row-space, and the pivot partitions.

The generalization of rooks problem and quasi-determinant of matrices

The rooks problem on a rectangular board is studied in this paper. Inspiring from the rook polynomials, the concept of determinant of square matrices is extended suitably to non-square matrices. We also provide some applications for quasi-determinant of non-square matrices. Furthermore, our algorithms for the rooks problem on a rectangular board and the quasi-determinant of non-square matrices are presented.

Rook polynomials in three and higher dimensions

The rook polynomial of a board counts the number of ways of placing nonattacking rooks on the board. In this paper, we describe how the properties of the two-dimensional rook polynomials generalize to the rook polynomials of “boards” in three and higher dimensions. We also define families of three-dimensional boards which generalize the two-dimensional triangle boards and the boards representing the probleme des rencontres. The rook coefficients of these threedimensional boards are shown to be related to famous number sequences such as the central factorial numbers, the number of Latin rectangles and the Genocchi numbers.

The ${1/k}$-Eulerian Polynomials

We use the theory of lecture hall partitions to define a generalization of the Eulerian polynomials, for each positive integer $k$. We show that these ${1}/{k}$-Eulerian polynomials have a simple combinatorial interpretation in terms of a single statistic on generalized inversion sequences. The theory provides a geometric realization of the polynomials as the $h^*$-polynomials of $k$-lecture hall polytopes. Many of the defining relations of the Eulerian polynomials have natural ${1}/{k}$-generalizations. In fact, these properties extend to a bivariate generalization obtained by replacing ${1}/{k}$ by a continuous variable. The bivariate polynomials have appeared in the work of Carlitz, Dillon, and Roselle on Eulerian numbers of higher order and, more recently, in the theory of rook polynomials.

Integrals Don’t Have Anything to Do with Discrete Math, Do They?

SummaryTo students just beginning their study of mathematics, the discipline appears to come in two distinct flavours: continuous and discrete. This article attempts to bridge the apparent divide by describing a surprising connection between these ostensible opposites. Various inhabitants from both worlds make appearances: rook polynomials, Euler's gamma function, derangements, and the Gaussian density. Uncloaking combinatorial proof of an integral identity serves as a thread tying these notions together.

The general $ \Gamma -$ compatible rook length polynomials

Rook placements and rook polynomials have been studied by mathematicians since the early 1970's. Since then many relationships between rook placements and other subjects have been discovered (cf. [1], [6-15]). In [2] and [3], K. Ding introduced the rook length polynomials and the $ \gamma - $compatible rook length polynomials. In [3] and [4], he used these polynomials to establish a connection between rook placements and algebraic geometry for the first time. In this paper, we give explicit formulas for the $ \gamma - $compatible rook length polynomials in more general cases than considered in [3]. In particular, we generalize the formula for the rook length polynomial in the parabolic case in [2] to the $ \gamma -$compatible rook length polynomial.

H-POLYNOMIALS AND ROOK POLYNOMIALS

The purpose of this paper is twofold. First we describe a useful procedure for computing the H-polynomials of reductive monoids. Second we use this procedure to compute the H-polynomial of the monoid of n × n matrices in terms of the q-analogues of the rook polynomials of Garsia and Remmel.

Staircase rook polynomials and Cayley’s game of Mousetrap

First introduced by Arthur Cayley in the 1850’s, the game of Mousetrap involves removing cards from a deck according to a certain rule. In this paper we find the rook polynomial for the number of Mousetrap decks in which at least two specified cards are removed. We also find a new expression for the rook polynomial for the number of decks in which exactly one specified card is removed and give expressions for counts of two kinds of Mousetrap decks in terms of other known combinatorial numbers.

Augmented Rook Boards and General Product Formulas

There are a number of so-called factorization theorems for rook polynomials that have appeared in the literature. For example, Goldman, Joichi and White showed that for any Ferrers board $B = F(b_1, b_2, \ldots, b_n)$, $$\prod_{i=1}^n (x+b_i-(i-1)) = \sum_{k=0}^n r_k(B) (x)\downarrow_{n-k}$$ where $r_k(B)$ is the $k$-th rook number of $B$ and $(x)\downarrow_k = x(x-1) \cdots (x-(k-1))$ is the usual falling factorial polynomial. Similar formulas where $r_k(B)$ is replaced by some appropriate generalization of the $k$-th rook number and $(x)\downarrow_k$ is replaced by polynomials like $(x)\uparrow_{k,j} = x(x+j) \cdots (x+j(k-1))$ or $(x)\downarrow_{k,j} = x(x-j) \cdots (x-j(k-1))$ can be found in the work of Goldman and Haglund, Remmel and Wachs, Haglund and Remmel, and Briggs and Remmel. We shall refer to such formulas as product formulas. The main goal of this paper is to develop a new rook theory setting in which we can give a uniform combinatorial proof of a general product formula that includes, as special cases, essentially all the product formulas referred to above. We shall also prove $q$-analogues and $(p,q)$-analogues of our general product formula.

A combinatorial interpretation of the [formula omitted]-hit numbers

Briggs and Remmel [K.S. Briggs, J.B. Remmel, A p , q -analogue of a formula of Frobenius, Electron. J. Combin. 10 (1) (2003) #R9] defined a p , q -analogue of the hit numbers and showed that they are polynomials in p and q with nonnegative coefficients for all Ferrers boards. Here we show that there is a natural extension of Dworkin’s statistic ξ [M. Dworkin, An interpretation for Garsia and Remmel’s q -hit numbers, J. Combin. Theory Ser. A 81 (1998) 149–175] so that for Ferrers boards, the p , q -hit numbers introduced by Briggs and Remmel arise by p , q -counting placements of n nonattacking rooks in the n × n board. Our proofs are based on different methods than those used by either Dworkin or Haglund [J. Haglund, q -Rook polynomials and matrices over finite fields, Adv. in Appl. Math. 20 (1998) 450–487]. Our proofs are based on showing that our combinatorial interpretations of the p , q -hit numbers satisfy certain recursions introduced by Briggs and Remmel.

Rook polynomials on two-dimensional surfaces and graceful labellings of graphs

By a simple trick we may generalise the rook polynomial for an n × n chessboard to various two-dimensional surfaces, the conventional chessboard corresponding to the torus. In the case of the Möbius band and the Klein bottle, there is a close connection to graceful labellings of graphs.

Rook polynomials to and from permanents

In this paper, we find an expression of the rook vector of a matrix A (not necessarily square) in terms of permanents of some matrices associated with A, and obtain some simple exact formulas for the permanents of all n× n Toeplitz band matrices of zeros and ones whose bands are of width not less than n−1.

Rook Theory for Perfect Matchings

In classical rook theory there is a fundamental relationship between the rook numbers and the hit numbers relative to any board. In that theory the k-th hit number of a board B can be interpreted as the number of permutations whose intersection with B is of size k. In the case of Ferrers boards there are q-analogues of the hit numbers and the rook numbers developed by A. M. Garsia and J. B. Remmel (1986, J. Combin. Theory, Ser. A41, 246–275) M. Dworkin (1996, “Generalizations of Rook Polynomials,” Ph. D. Thesis, Brandeis University”; 1998, J. Combin. Theory, Ser. A81, 149–175) and J. Haglund (1998, Adv. Appl. Math.20, 450–487). In this paper we develop a rook theory appropriate for shifted partitions, where hit numbers can be interpreted as the number of perfect matchings in the complete graph whose intersection with the board is of size k. We show there is also analogous q-theory for the rook and hit numbers for these shifted Ferrers boards.

Further Investigations Involving Rook Polynomials With Only Real Zeros

We study the zeros of two families of polynomials related to rook theory and matchings in graphs. One of these families is based on the cover polynomial of a digraph introduced by Chung and Graham . Another involves a version of the ‘hit polynomial’ of rook theory, but which applies to weighted matchings in (non-bipartite) graphs. For both of these families we prove a result which is analogous to a theorem of the author, K. Ono, and D. G. Wagner, namely that for Ferrers boards the hit polynomial has only real zeros. We also show that for each of these families there is a general conjecture involving arrays of numbers satisfying inequalities which contains these theorems as special cases. We provide evidence for the truth of these conjectures by proving other special cases and discussing computational experiments.

Generalized Rook Polynomials

Generalizing the notion of placing rooks on a Ferrers board leads to a new class of combinatorial models and a new class of rook polynomials. Connections are established with absolute Stirling numbers and permutations, Bessel polynomials, matchings, multiset permutations, hypergeometric functions, Abel polynomials and forests, and polynomial sequences of binomial type. Factorization and reciprocity theorems are proved and a q-analogue is given.

Lucas numbers and the menage problem

A solution to the menage problem is given through the use of largest rook polynomials. These are rook polynomials whose coefficients have greatest sums which turn out to be Fibonacci numbers, while the sums of the coefficients of the rook polynomial for the reduced ménage problem are Lucas numbers.

Maximising the Permanent of (0,1)-Matrices and the Number of Extensions of Latin Rectangles

Let $k\geq2$, $m\geq5$ and $n=mk$ be integers. By finding bounds for certain rook polynomials, we identify the $k\times n$ Latin rectangles with the most extensions to $(k+1)\times n$ Latin rectangles. Equivalently, we find the $(n-k)$-regular subgraphs of $K_{n,n}$ 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. Without the restriction on $n$ being a multiple of $k$ we solve the above problem (and the corresponding minimisation problem) for $k=2$. We also provide some computational results for small values of $n$ and $k$. Our results partially settle two open problems of Minc and conjectures by Merriell, and Godsil and McKay.

Dual rook polynomials

A theorem contained in the paper ‘A combinatoric formula’ by Wang, Lee and Tan (J. Math. Anal. Appl. 160 (1991) 500–503) gives rise to the definition of certain polynomials associated with boards: Stimulated by the analogy to the well-known rook polynomials, we call them ‘dual rook polynomials’. We show that in the cases of Ferrers boards and skew boards the evaluation of these polynomials at −1 always yields values −1, 0 or 1, generalizing the theorem cited above. Moreover, we evaluate these polynomials at −2. Finally, we state three conjectures that are quite well supported by empirical tests: Two of these conjectures are known to be true for rook polynomials.

Rook Theory and Hypergeometric Series

The number of ways of placingknon-attacking rooks on a Ferrers board is expressed as a hypergeometric series of a type originally studied by Karlsson (J. Math. Phys.12(1971), 270–271) and Minton (J. Math. Phys.11(1970), 1375–1376). Known transformation identities for series of this type translate into new theorems about rook polynomials.