Counting coloured planar maps: differential equations

Olivier Bernardi (1) , Mireille Bousquet-Mélou (2)

1 Brandeis University
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

We address the enumeration of q-coloured planar maps counted by the number of edges and the number of monochromatic edges. We prove that the associated generating function is differentially algebraic, that is, satisfies a non-trivial polynomial differential equation with respect to the edge variable. We give explicitly a differential system that characterizes this series. We then prove a similar result for planar triangulations, thus generalizing a result of Tutte dealing with their proper q-colourings. In statistical physics terms, we solve the q-state Potts model on random planar lattices. This work follows a first paper by the same authors, where the generating function was proved to be algebraic for certain values of q, including q=1, 2 and 3. It is known to be transcendental in general. In contrast, our differential system holds for an indeterminate q. For certain special cases of combinatorial interest (four colours; proper q-colourings; maps equipped with a spanning forest), we derive from this system, in the case of triangulations, an explicit differential equation of order 2 defining the generating function. For general planar maps, we also obtain a differential equation of order 3 for the four-colour case and for the self-dual Potts model.

Mots clés: Differentially algebraic series; Coloured planar maps; Tutte polynomial; Enumeration

Domaines: Combinatoire [math.CO]

Olivier Bernardi, Mireille Bousquet-Mélou. Counting coloured planar maps: differential equations. Communications in Mathematical Physics, 2017, 354 (1), pp.31-84. ⟨hal-01174130v2⟩