Mathematics – Combinatorics
Scientific paper
2010-10-27
Mathematics
Combinatorics
Scientific paper
A linear chord diagram canonically determines a fatgraph and hence has an associated genus $g$. We compute the natural generating function ${\bf C}_g(z)=\sum_{n\geq 0} {\bf c}_g(n)z^n$ for the number ${\bf c}_g(n)$ of linear chord diagrams of fixed genus $g\geq 1$ with a given number $n\geq 0$ of chords and find the remarkably simple formula ${\bf C}_g(z)=z^{2g}R_g(z) (1-4z)^{{1\over 2}-3g}$, where $R_g(z)$ is a polynomial of degree at most $g-1$ with integral coefficients satisfying $R_g({1\over 4})\neq 0$ and $R_g(0) = {\bf c}_g(2g)\neq 0.$ In particular, ${\bf C}_g(z)$ is algebraic over $\mathbb C(z)$, which generalizes the corresponding classical fact for the generating function ${\bf C}_0(z)$ of the Catalan numbers. As a corollary, we also calculate a related generating function germaine to the enumeration of knotted RNA secondary structures, which is again found to be algebraic.
Andersen Jørgen Ellegaard
Penner Robert C.
Reidys Christian M.
Waterman M. S.
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