Dennis E. White

The cyclic sieving phenomenon

The cyclic sieving phenomenon is defined for generating functions of a set affording a cyclic group action, generalizing Stembridges q = -1 phenomenon. The phenomenon is shown to appear in various situations, involving q-binomial coefficients, Polya-Redfield theory, polygon dissections, noncrossing partitions, finite reflection groups, and some finite field q-analogues.

p,q -Stirling numbers and set partition statistics

Abstract We give bijections on restricted growth functions and rook placements on stairstep Ferrers boards to show that the q -Stirling numbers of the second kind, described by Gould, arise as generating functions for various statistics on set partitions. We also describe a two-variable, p, q -Stirling number which is the generating function for the joint distribution of pairs of statistics.

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.

Combinatorial Gray Codes

We consider families

Some connections between the Littlewood-Richardson rule and the construction of Schensted

\{ {\bf C}(n,k):O \leqq k \leqq n\}

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

where each

Sign-Balanced Posets

{\bf C}(n,k)

Interpolating set partition statistics

is a set of combinatorial objects,

The combinatorics of q -Charlier polynomials

C(n,k) = |{\bf C}(n,k)|

ROOK THEORY. II: BOARDS OF BINOMIAL TYPE

satisfies a recursion