8 research outputs found
{\Gamma}-species and the enumeration of k-trees
We study the class of graphs known as k-trees through the lens of Joyal's
theory of combinatorial species (and an equivariant extension known as
'-species' which incorporates data about 'structural' group actions).
This culminates in a system of recursive functional equations giving the
generating function for unlabeled k-trees which allows for fast, efficient
computation of their numbers. Enumerations up to k = 10 and n = 30 (for a
k-tree with (n+k-1) vertices) are included in tables, and Sage code for the
general computation is included in an appendix.Comment: 26 pages; includes Python cod
Geometric combinatorics and computational molecular biology: branching polytopes for RNA sequences
Questions in computational molecular biology generate various discrete
optimization problems, such as DNA sequence alignment and RNA secondary
structure prediction. However, the optimal solutions are fundamentally
dependent on the parameters used in the objective functions. The goal of a
parametric analysis is to elucidate such dependencies, especially as they
pertain to the accuracy and robustness of the optimal solutions. Techniques
from geometric combinatorics, including polytopes and their normal fans, have
been used previously to give parametric analyses of simple models for DNA
sequence alignment and RNA branching configurations. Here, we present a new
computational framework, and proof-of-principle results, which give the first
complete parametric analysis of the branching portion of the nearest neighbor
thermodynamic model for secondary structure prediction for real RNA sequences.Comment: 17 pages, 8 figure
Counting unlabeled k-trees
Abstract We count unlabeled k-trees by properly coloring them in k + 1 colors and then counting orbits of these colorings under the action of the symmetric group on the colors