Enumerating partitions arising in homotopy theory

Some Problems Arising from Homotopy-theoretic Methods in Knot Theory

This is a partial list of some interesting questions that arose in the past decade or so from applications of homotopy-theoretic methods in knot and link theory. The ones I have in mind (because those are the ones I am familiar with) are manifold calculus of functors, cosimplicial spaces, and operad actions. (Some problems are also about configuration space integrals as they have proven to be a...

Enumerating Consecutive and Nested Partitions for Graphs

Consecutive and nested partitions have been extensively studied in the set-partition problem as tools with which to search efficiently for an optimal partition. We extend the study of consecutive and nested partitions on a set of integers to the vertex-set of a graph. A subset of vertices is considered consecutive if the subgraph induced by the subset is connected. In this sense the partition p...

Homotopy Theory of Homotopy Algebras

The goal of this paper is to study the homotopy theory of homotopy algebras over a Koszul operad with their infinity morphisms. The method consists in endowing the category of coalgebras over the Koszul dual cooperad with a model category structure. CONTENTS Introduction 1 1. Recollections 2 2. Model category structure for coalgebras 5 3. Conclusion 14 Appendix A. A technical lemma 15 Reference...

Enumerating finite set partitions according to the number of connectors

Let P (n, k) denote the set of partitions of [n] = {1, 2, . . . , n} containing exactly k blocks. Given a partition Π = B1/B2/ · · · /Bk ∈ P (n, k) in which the blocks are listed in increasing order of their least elements, let π = π1π2 · · · πn denote the canonical sequential form wherein j ∈ Bπj for all j ∈ [n]. In this paper, we supply an explicit formula for the generating function which co...

Configurations in Homotopy Theory