Bijective Enumeration of 3-Factorizations of an N-Cycle

This paper is dedicated to the factorizations of the symmetric group. Introducing a new bijection for partitioned 3-cacti, we derive an el- egant formula for the number of factorizations of a long cycle into a product of three permutations. As the most salient aspect, our construction provides the first purely combinatorial computation of this number

Structure and enumeration of (3+1)-free posets

A poset is (3+1)-free if it does not contain the disjoint union of chains of length 3 and 1 as an induced subposet. These posets play a central role in the (3+1)-free conjecture of Stanley and Stembridge. Lewis and Zhang have enumerated (3+1)-free posets in the graded case by decomposing them into bipartite graphs, but until now the general enumeration problem has remained open. We give a finer decomposition into bipartite graphs which applies to all (3+1)-free posets and obtain generating functions which count (3+1)-free posets with labelled or unlabelled vertices. Using this decomposition, we obtain a decomposition of the automorphism group and asymptotics for the number of (3+1)-free posets.Comment: 28 pages, 5 figures. New version includes substantial changes to clarify the construction of skeleta and the enumeration. An extended abstract of this paper appears as arXiv:1212.535

The complexity of the normal surface solution space

Normal surface theory is a central tool in algorithmic three-dimensional topology, and the enumeration of vertex normal surfaces is the computational bottleneck in many important algorithms. However, it is not well understood how the number of such surfaces grows in relation to the size of the underlying triangulation. Here we address this problem in both theory and practice. In theory, we tighten the exponential upper bound substantially; furthermore, we construct pathological triangulations that prove an exponential bound to be unavoidable. In practice, we undertake a comprehensive analysis of millions of triangulations and find that in general the number of vertex normal surfaces is remarkably small, with strong evidence that our pathological triangulations may in fact be the worst case scenarios. This analysis is the first of its kind, and the striking behaviour that we observe has important implications for the feasibility of topological algorithms in three dimensions.Comment: Extended abstract (i.e., conference-style), 14 pages, 8 figures, 2 tables; v2: added minor clarification

Chain enumeration of kk-divisible noncrossing partitions of classical types

We give combinatorial proofs of the formulas for the number of multichains in the $k$-divisible noncrossing partitions of classical types with certain conditions on the rank and the block size due to Krattenthaler and M{\"u}ller. We also prove Armstrong's conjecture on the zeta polynomial of the poset of $k$-divisible noncrossing partitions of type $A$ invariant under a $180^\circ$ rotation in the cyclic representation.Comment: 23 pages, 9 figures, final versio