The generating function of planar Eulerian orientations
Abstract
The enumeration of planar maps equipped with an Eulerian orientation has attracted attention in both combinatorics and theoretical physics since at least 2000. The case of 4valent maps is particularly interesting: these orientations are in bijection with properly 3coloured quadrangulations, while in physics they correspond to configurations of the ice model. We solve both problems  namely the enumeration of planar Eulerian orientations and of 4valent planar Eulerian orientations  by expressing the associated generating functions as the inverses (for the composition of series) of simple hypergeometric series. Using these expressions, we derive the asymptotic behaviour of the number of planar Eulerian orientations, thus proving earlier predictions of Kostov, ZinnJustin, Elvey Price and Guttmann. This behaviour, $\mu^n /(n \log n)^2$, prevents the associated generating functions from being Dfinite. Still, these generating functions are differentially algebraic, as they satisfy nonlinear differential equations of order $2$. Differential algebraicity has recently been proved for other map problems, in particular for maps equipped with a Potts model. Our solutions mix recursive and bijective ingredients. In particular, a preliminary bijection transforms our oriented maps into maps carrying a height function on their vertices. In the 4valent case, we also observe an unexpected connection with the enumeration of maps equipped with a spanning tree that is internally inactive in the sense of Tutte. This connection remains to be explained combinatorially.
 Publication:

arXiv eprints
 Pub Date:
 March 2018
 arXiv:
 arXiv:1803.08265
 Bibcode:
 2018arXiv180308265B
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 37 pp