Significance of Knotted Structures for Function of Proteins and Nucleic Acids - September 17-21, 2014

Significance of Knotted Structures for Function of Proteins and Nucleic Acids

Poster Session I

24 – POS Board 24 A Combinatorial Interpretation of the

Coefficients

Thomas J. X. Li , Christian M. Reidys Institut for Matematik og Datalogi, University of Southern Denmark, Denmark

Unicellular map and its shape have been applied to RNA pseudoknotted structure filtered by its topological genus. Studying the virtual Euler characteristic of the moduli space of curves, Harer and Zagier compute the generating function of unicellular maps of genus g . They furthermore identify coefficients, , which fully determine the series . The main result of this abstract is a combinatorial interpretation of . We show that these enumerate a class of unicellular maps, which correspond 1-to- to a specific type of trees, referred to as O-trees, see Figure 1. We show how to generate from this specific class of O-trees to the class of shapes, see Figure 2. We prove the are positive integers that satisfy a two term recursion

We furthermore prove that for any fixed g , the sequence

is log-concave, where

,

for . Keywords : unicellular map, fatgraph, O-tree, shape-polynomial, recursion

Figure 1. Figure 2.

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