Enumeration of linear chord diagrams

2010/10/27 by Andersen, J. E., Penner, R. C., Reidys, C. M. +1
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1010.5614

Abstract

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)=∑_{n≥ 0} \bf c_g(n)zⁿ for the number \bf c_g(n) of linear chord diagrams of fixed genus g≥ 1 with a given number n≥ 0 of chords and find the remarkably simple formula \bf C_g(z)=z^2gR_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)≠ 0 and R_g(0) = \bf c_g(2g)≠ 0. In particular, \bf C_g(z) is algebraic over \mathbb C(z), which generalizes the corresponding classical fact for the generating function \bf C₀(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.

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