We prove that the set of limit groups is recursively enumerable, answering a question of Delzant. One ingredient of the proof is the observation that a finitely presented group with local retractions (à la Long and Reid) is coherent and, furthermore, there exists an algorithm that computes presentations for finitely generated subgroups. The other main ingredient is the ability to algorithmicall...

We consider the problem of enumerating periodic σ-juggling sequences of length n for multiplex juggling, where σ is the initial state (or landing schedule) of the balls. We first show that this problem is equivalent to choosing 1’s in a specified matrix to guarantee certain column and row sums, and then using this matrix, derive a recursion. This work is a generalization of earlier work of Fan ...

We consider packings of n identical spherical particles that bind to one another without deformation or overlap, and without any long range interactions. Combining graph theoretic enumeration with basic geometry, we analytically solve for packings of n ≤ 10 particles satisfying minimal rigidity constraints (≥ 3 contacts per particle and ≥ 3n− 6 total contacts). We find that the onset of packing...

This paper makes a contribution to the enumeration of trees. We prove a new result about k-way trees, point out some special cases, use it to give a new proof of Cayley’s enumeration formula for labelled trees, and observe that our techniques allow various types of k-way trees to be generated uniformly at random. We recall the definition: a k-way tree is either empty or it consists of a root no...

We describe a new algebraic technique for enumerating self-avoiding walks on the rectangular lattice. The computational complexity of enumerating walks of N steps is of order 3 N/4 times a polynomial in N , and so the approach is greatly superior to direct counting techniques. We have enumerated walks of up to 39 steps. As a consequence, we are able to accurately estimate the critical point, cr...

We consider the problem of enumerating permutations with exactly r occurrences of the pattern 1324 and derive functional equations for this general case as well as for the pattern avoidance (r = 0) case. The functional equations lead to a new algorithm for enumerating length n permutations that avoid 1324. This approach is used to enumerate the 1324-avoiders up to n = 31. We also extend those f...

Enumerating formulae are constructed which count the number of partitions of a positive integer into positive summands in arithmetic progression with common difference D. These enumerating formulae (denoted pD(n)) which are given in terms of elementary divisor functions together with auxiliary arithmetic functions (to be defined) are then used to establish a known characterisation for an intege...

A bimatrix game (A, B) is called a game of rank k if the rank of the matrix A + B is at most k. We consider the problem of enumerating the Nash equilibria in (non-degenerate) games of rank 1. In particular, we show that even for games of rank 1 not all equilibria can be reached by a Lemke– Howson path and present a parametric simplex-type algorithm for enumerating all Nash equilibria of a non-d...