Rook Theory Research Articles

Rook theory of the Etzion-Silberstein Conjecture

Rook theory of the Etzion-Silberstein Conjecture

Rook Theory of the Finite General Linear Group

ABSTRACTMatrices over a finite field having fixed rank and restricted support are a natural q-analog of rook placements on a board, even though their enumeration is known to yield nonpolynomial answers in some cases. We develop this q-rook theory by defining a corresponding analog 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-analogs of rook theory, as well as settling a polynomiality conjecture and finding experimentally a counterexample of a positivity conjecture of the authors and Klein.

Elliptic extensions of the alpha-parameter model and the rook model for matchings

We construct elliptic extensions of the alpha-parameter rook model introduced by Goldman and Haglund and of the rook model for matchings of Haglund and Remmel. In particular, we extend the product formulas of these models to the elliptic setting. By specializing the parameter α in our elliptic extension of the alpha-parameter model and the shape of the Ferrers board in different ways, we obtain elliptic analogues of the Stirling numbers of the first kind and of the Abel polynomials, and an a,q-analogue of the matching numbers. We further generalize the rook theory model for matchings by introducing l-lazy graphs which correspond to l-shifted boards, where l is a finite vector of positive integers. The corresponding elliptic product formula generalizes Haglund and Remmel's product formula for matchings already in the non-elliptic basic case.

An elliptic extension of the general product formula for augmented rook boards

Rook theory has been investigated by many people since its introduction by Kaplansky and Riordan in 1946. Goldman, Joichi, and White in 1975 showed that the sum over k of the product of the (n−k)th rook numbers multiplied by the kth falling factorial polynomials factorize into a product. In the sequel, different types of generalizations and analogues of this product formula have been derived by various authors. In 2008, Miceli and Remmel constructed a rook theory model involving augmented rook boards in which they showed the validity of a general product formula which can be specialized to all other product formulas that so far have appeared in the literature on rook theory. In this work, we construct an elliptic extension of the q-analogue of Miceli and Remmel’s result. Special cases yield elliptic extensions of various known rook theory models.

Some combinatorial identities involving noncommuting variables

We derive combinatorial identities for variables satisfying specific sets of commutation relations. The identities thus obtained extend corresponding ones for $q$-commuting variables $x$ and $y$ satisfying $yx=qxy$. In particular, we obtain weight-dependent binomial theorems, functional equations for generalized exponential functions, we propose a derivative of noncommuting variables, and finally utilize one of the considered weight functions to extend rook theory. This leads us to an extension of the $q$-Stirling numbers of the second kind, and of the $q$-Lah numbers. Nous obtenons des identités combinatoires pour des variables satisfaisant des ensembles spécifiques de relations de commutation. Ces identités ainsi obtenues généralisent leurs analogues pour des variables $q$-commutantes $x$ et $y$ satisfaisant $yx=qxy$. En particulier, nous obtenons des théorèmes binomiaux dépendant du poids, des équations fonctionnelles pour les fonctions exponentielles généralisées, nous proposons une dérivée des variables non-commutatives, et finalement nous utilisons l’une des fonctions de poids considérées pour étendre la théorie des tours. Nous en déduisons une généralisation des $q$-nombres de Stirling de seconde espèce et des $q$-nombres de Lah.

Partitions, Rooks, and Symmetric Functions in Noncommuting Variables

Let $\Pi_n$ denote the set of all set partitions of $\{1,2,\ldots,n\}$. We consider two subsets of $\Pi_n$, one connected to rook theory and one associated with symmetric functions in noncommuting variables. Let ${\cal E}_n\subseteq\Pi_n$ be the subset of all partitions corresponding to an extendable rook (placement) on the upper-triangular board, ${\cal T}_{n-1}$. Given $\pi\in\Pi_m$ and $\sigma\in\Pi_n$, define their slash product to be $\pi|\sigma=\pi\cup(\sigma+m)\in\Pi_{m+n}$ where $\sigma+m$ is the partition obtained by adding $m$ to every element of every block of $\sigma$. Call $\tau$ atomic if it can not be written as a nontrivial slash product and let ${\cal A}_n\subseteq\Pi_n$ denote the subset of atomic partitions. Atomic partitions were first defined by Bergeron, Hohlweg, Rosas, and Zabrocki during their study of $NCSym$, the symmetric functions in noncommuting variables. We show that, despite their very different definitions, ${\cal E}_n={\cal A}_n$ for all $n\ge0$. Furthermore, we put an algebra structure on the formal vector space generated by all rook placements on upper triangular boards which makes it isomorphic to $NCSym$. We end with some remarks.

Rook-by-rook rook theory: Bijective proofs of rook and hit equivalences

Suppose μ and ν are integer partitions of n, and N > n . It is well known that the Ferrers boards associated to μ and ν are rook-equivalent iff the multisets [ μ i + i : 1 ⩽ i ⩽ N ] and [ ν i + i : 1 ⩽ i ⩽ N ] are equal. We use the Garsia–Milne involution principle to produce a bijective proof of this theorem in which non-attacking rook placements for μ are explicitly matched with corresponding placements for ν. One byproduct is a direct combinatorial proof that the matrix of Stirling numbers of the first kind is the inverse of the matrix of Stirling numbers of the second kind. We also prove q-analogues and p , q -analogues of these results. We also use the Garsia–Milne involution principle to show that for any two rook boards B and B ′ , if B and B ′ are bijectively rook-equivalent, then B and B ′ are bijectively hit-equivalent.

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.

M-Rook numbers and a generalization of a formula of Frobenius to [formula omitted

In ordinary rook theory, rook placements are associated to permutations of the symmetric group S n . We provide a generalization of this theory in which “ m-rook placements” are related to elements of C m ≀ S n , where C m is the cyclic group of order m. Within this model, we define and interpret combinatorially a p , q -analogue of the m-rook numbers. We also define a p , q -analogue of the m-hit numbers and show that the coefficients of these polynomials in p and q are nonnegative integers for m-Ferrers boards. Finally, we define statistics des m ( σ ) , maj m ( σ ) , and comaj m ( σ ) as analogues of the ordinary descent, major, and comajor index statistics and prove a generalization of a formula of Frobenius that relates these statistics to generalized p , q -Stirling numbers of the second kind.

Rook Theory, Generalized Stirling Numbers and $(p,q)$-Analogues

In this paper, we define two natural $(p,q)$-analogues of the generalized Stirling numbers of the first and second kind $S^1(\alpha,\beta,r)$ and $S^2(\alpha,\beta,r)$ as introduced by Hsu and Shiue [Adv. in Appl. Math. 20 (1998), 366–384]. We show that in the case where $\beta =0$ and $\alpha$ and $r$ are nonnegative integers both of our $(p,q)$-analogues have natural interpretations in terms of rook theory and derive a number of generating functions for them. We also show how our $(p,q)$-analogues of the generalized Stirling numbers of the second kind can be interpreted in terms of colored set partitions and colored restricted growth functions. Finally we show that our $(p,q)$-analogues of the generalized Stirling numbers of the first kind can be interpreted in terms of colored permutations and how they can be related to generating functions of permutations and signed permutations according to certain natural statistics.

Rook theory and cycle-counting permutation statistics

A statistic is found to combinatorially generate the cycle-counting q-hit numbers, defined algebraically by Haglund [Adv. in Appl. Math. 17 (1996) 408–459]. We then define the notion of a cycle-Mahonian pair of statistics (generalizing that of a Mahonian statistic), and show that our newly discovered statistic is part of such a pair. Finally, we note a second example of a cycle-Mahonian pair of statistics which leads us to define the stronger property of being a cycle-Euler–Mahonian pair.

Cycles and perfect matchings

Fan Chung and Ron Graham (J. Combin. Theory Ser. B 65 (1995) 273–290) introduced the cover polynomial for a directed graph and showed that it was connected with classical rook theory. Dworkin (J. Combin. Theory Ser. B 71 (1997) 17–53) showed that the cover polynomial naturally factors for directed graphs associated with Ferrers boards. The authors (Adv. Appl. Math. 27 (2001) 438–481) developed a rook theory for shifted Ferrers boards where the analogue of a rook placement is replaced by a partial perfect matching of K 2 n , the complete graph on 2 n vertices. In this paper, we show that an analogue of Dworkin's result holds for shifted Ferrers boards in this setting. We also show how cycle-counting matching numbers are connected to cycle-counting “hit numbers” (which involve perfect matchings of K 2 n ).

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.

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.

Rook theory, compositions, and zeta functions

A new family of Dirichlet series having interesting combinatorial properties is introduced. Although they have no functional equation or Euler product, under the Riemann Hypothesis it is shown that these functions have no zeros in R e ( s ) > 1 / 2 \mathrm {Re}(s)>1/2 . Some identities in the ring of formal power series involving rook theory and continued fractions are developed.

Partial ferrers matrices

We introduce a class ofn×n matrices of 0’s and 1’s which can be regarded as generalizations of the classical Ferrers boards of rook theory. Assuming the matrices are fully indecomposable, we determine the minimum permanent and the minimum number of 1’s as a function ofn. We also characterize these matrices in terms of weighted, top-rooted trees.

Rook theory III. Rook polynomials and the chromatic structure of graphs

This paper studies the relationship between the rook vector of a general board and the chromatic structure of an associated set of graphs. We prove that every rook vector is a chromatic vector. We give algebraic relations between the factorial polynomials of two boards and their union and sum, and the chromatic polynomials of two graphs and their union and sum.

Rook Theory—IV. Orthogonal Sequences of Rook Polynomials

We explain combinatorially the occurrence of certain classical sequences of orthogonal polynomials as sequences of rook polynomials, and we give some new examples related to general stairstep boards.

Rook Theory. II: Boards of Binomial Type

We explore the relation between rook theory of Ferrers boards and polynomial sequences of binomial type. Recursion and explicit formulas for the rook numbers of Riordan’s trapezoidal boards are derived. It is shown that some classic sequences of rook and factorial polynomials are of binomial type. A class of boards corresponding to Abel’s theorem and its generalizations are constructed. The results are developed from both algebraic and combinatorial viewpoints.