Jay R. Goldman

Rook theory. I. Rook equivalence of Ferrers boards

We introduce a new tool, the factorial polynomials, to study rook equivalence of Ferrers boards. We provide a set of invariants for rook equivalence as well as a very simple algorithm for deciding rook equivalence of Ferrers boards. We then count the number of Ferrers boards rook equivalent to a given Ferrers board. Introduction. Let N denote the set of positive integers. A board is a finite subset of N x N. Intuitively, a board is an array of squares or cells arranged in rows and columns, i.e., a board is a subset of the set of squares of an n x n chessboard. We shall frequently utilize this intuitive terminology. We consider two boards to be the same board if one is a translate of the other, i.e., boards B and B are the same if there exist integers a and b such that B {(i + a, j + b): (i, j) € B\. Thus, only the geometric configuration or the relative positions of the cells is of significance. For a board B, \B\ denotes the number of cells in B, Let r, be the number of ways of placing k nontaking rooks (no two in the same row or column) on the board B, i.e., the number of /^-subsets of the set B such that no two elements of a ^-subset have the same first component or the same second component. When no confusion can arise we suppress the B and write r, . The rook vector of a board B is defined to be the vector r(B) = (rQ, rj, r2, . . . ) where rQ = 1. Note that from some point on all the r.s are zero, in particular r. = 0 for i > \B\. Two boards are called rook equivalent if they have the same rook vector. A board B is a Ferrers board if there exists a nondecreasing finite sequence of positive integers h j, h2, . . . , h such that B = {(i, /): i < c and j < h.\. Intuitively, a Ferrers board is a board made up of adjacent solid columns of cells with a common lower edge and such that the height of the columns from left to right forms a nondecreasing sequence. If the heights of the columns form a strictly increasing sequence, then we call the board an increasing Ferrers board. Examples of a Ferrers board and an increasing Ferrers board are given in Figures 1(a) and 1(b), respectively. Received by the editors August 21, 1974. AMS (MOS) subject classifications (1970). Primary 05A10, 05A15, 05A19.

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

Abstract 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.

Formal languages and enumeration

Abstract We present a theory of generating functions in countably many non-commuting variables. This generalizes the theory of context free languages. Applications are given to compositions of a number, rooted planar tree, dissected polygons, and the theory of simple random walks.

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.

Rook theory. V rook polynomials, Möbius inversion and the umbral calculus

Abstract In this paper we provide the first general expressions for the rook and factorial polynomials associated with a general board. We use Mobius inversion over the lattice of partitions of a set to provide explicit formulas. Alternate formulas are given by the methods of the umbral calculus. We also state a general conjecture concerning the roots of rook polynomials and prove the conjecture for a special class of boards.

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.

Numbers of solutions of congruences: Poincare series for algebraic curves

Abstract Let p be a fixed prime; f(x1,…,xk) a polynomial over Z p, the p-adic integers; cn the number of solutions in Z /pn Z ; and Pf(t) = Σciti the Poincare series. An elementary algebraic-combinatorial approach is used to study the cn and pf(t) for algebraic curves all of whose singularities are “locally” of the form αxa = βyb. Polynomials in one variable are discussed as a corollary to the treatment of algebraic curves.

Hurwitz sequences, the Farey process, and general continued fractions

Etude des fractions continues unitaires [b 0 ; a 1 /b 1 , a 2 /b 2 , …] ou ai et bi sont des entiers, bi>0 pour tout i>0 et |a i |=1 pour tout i. Relation entre toutes les fractions continues unitaires convergeant vers une limite α