A combinatorial construction for two formulas in Slater’s list

We set up a combinatorial framework for inclusion-exclusion on the partitions into distinct parts to obtain an alternative generating function of partitions into distinct and non-consecutive parts. In connection with Rogers–Ramanujan identities, the generating function yields two formulas in Slater’s list. The same formulas were constructed by Hirschhorn. Similar formulas were obtained by Bringmann, Mahlburg and Nataraj. We also use staircases to give alternative triple series for partitions into [Formula: see text]-distinct parts for any [Formula: see text].

Analytic combinatorics for bioinformatics I: seeding methods

Seeding heuristics are the most widely used strategies to speed up sequence alignment in bioinformatics. Such strategies are most successful if they are calibrated, so that the speed-versus-accuracy trade-off can be properly tuned. In the widely used case of read mapping, it has been so far impossible to predict the success rate of competing seeding strategies for lack of a theoretical framework. Here I present an approach to estimate such quantities based on the theory of analytic combinatorics. In a nutshell, the strategy is to specify a combinatorial construction of reads where the seeding heuristic fails, translate this specification into a generating function using formal rules, and finally extract the probabilities of interest from the singularities of the generating function. I use this approach to construct simple estimators of the success rate of the seeding heuristic under different types of sequencing errors. I also show how the analytic combinatorics strategy can be used to compute the associated type I and type II error rates (mapping the read to the wrong location, or being unable to map the read). Finally, I show how analytic combinatorics can be used to estimate average quantities such as the expected number of errors in reads where the seeding heuristic fails. Overall, this work introduces a theoretical and practical framework to find the success rate of seeding heuristics and related problems in bioinformatics.

Dijkgraaf–Witten type invariants of Seifert surfaces in 3-manifolds

We introduce defects, with internal gauge symmetries, on a knot and Seifert surface to a knot into the combinatorial construction of finite gauge-group Dijkgraaf–Witten theory. The appropriate initial data for the construction are certain three object categories, with coefficients satisfying a partially degenerate cocycle condition.

Planar algebras and the decategorification of bordered Khovanov homology

We give a simple, combinatorial construction of a unital, spherical, non-degenerate *-planar algebra over the ring [Formula: see text]. This planar algebra is similar in spirit to the Temperley–Lieb planar algebra, but computations show that they are different. The construction comes from the combinatorics of the decategorifications of the type A and type D structures in the author’s previous work on bordered Khovanov homology. In particular, the construction illustrates how gluing of tangles occurs in the bordered Khovanov homology and its difference from that in Khovanov’s tangle homology without being encumbered by any extra homological algebra. It also provides a simple framework for showing that these theories are not related through a simple process, thereby confirming recent work of Manion. Furthermore, using Khovanov’s conventions and a state sum approach to the Jones polynomial, we obtain new invariant for tangles in [Formula: see text] where [Formula: see text] is a compact, planar surface with boundary, and the tangle intersects each boundary cylinder in an even number of points. This construction naturally generalizes Khovanov’s approach to the Jones polynomial.