We introduce the concept of a segment of a degenerate convex polytope specified by a system of linear constraints, and explain its importance in developing algorithms for enumerating the faces. Using segments, we describe an algorithm that enumerates all the faces, in time polynomial in their number. The role of segments in the unsolved problem of enumerating the extreme points of a convex poly...

Recently, Aichholzer introduced the remarkable concept of the so-called triangulation path (of a triangulation with respect to a segment), which has the potential of providing efficient counting of triangulations of a point set, and efficient representations of all such triangulations. Experiments support such evidence, although – apart from the basic uniqueness properties – little has been pro...

In some of these boxes, the setter puts some of the digits 1–9; the aim of the solver is to complete the grid by filling in a digit in every box in such a way that each row, each column, and each 3× 3 box contains each of the digits 1–9 exactly once. In this note, we discuss the problem of enumerating all possible Sudoku grids. This is a very natural problem, but, perhaps surprisingly, it seems...

An (nk)-configuration is a set of n points and n lines in the projective plane such that their point – line incidence graph is k-regular. The configuration is geometric, topological, or combinatorial depending on whether lines are considered to be straight lines, pseudolines, or just combinatorial lines. We provide an algorithm for generating, for given n and k, all topological (nk)-configurati...

We consider the problem of enumerating polynomials over Fq, that have certain coefficients prescribed to given values and permute certain substructures of Fq. In particular, we are interested in the group of N -th roots of unity and in the submodules of Fq. We employ the techniques of Konyagin and Pappalardi to obtain results that are similar to their results in [Finite Fields and their Applica...

Counting the number of distinct colorings of various discrete objects, via Burnside’s Lemma and Pólya Counting, is a traditional problem in combinatorics. Motivated by a method for proving upper bounds on the order of the minimal recurrence relation satisfied by a set of tiling instances, we address a related problem in a more general setting. Given an m× n chessboard and a fixed set of (possib...

We study two enumeration problems for up-down alternating trees, i.e., rooted labelled trees T , where the labels v1, v2, v3, . . . on every path starting at the root of T satisfy v1 < v2 > v3 < v4 > · · · . First we consider various tree families of interest in combinatorics (such as unordered, ordered, d-ary and Motzkin trees) and study the number Tn of different up-down alternating labelled ...

Projective reflection groups have been recently defined by the second author. They include a special class of groups denoted G(r, p, s, n) which contains all classical Weyl groups and more generally all the complex reflection groups of type G(r, p, n). In this paper we define some statistics analogous to descent number and major index over the projective reflection groups G(r, p, s, n), and we ...

An arrangement of the multi-set {1,1,2,2, . . . , n, n} is said to be “split-pair” if for all i < n, between the two occurrences of i there is exactly one i + 1. We enumerate the number of split-pair arrangements and in particular show that the number of such arrangements is (−1)n+12n(22n − 1)B2n where Bi is the ith Bernoulli number. © 2007 Elsevier Inc. All rights reserved.

Groupoids satisfying the equation x(yz) = (xy)(xz) are called left distributive, or LD-groupoids. We give an algorithm for their enumeration and prove several results on the collection of LD-groupoids extending a given monounary algebra.