420 research outputs found
Counting coloured planar maps: differential equations
We address the enumeration of q-coloured planar maps counted bythe 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 withrespect to the edge variable.
We give explicitly a differential systemthat characterizes this series. We then
prove a similar result for planar triangulations, thus generalizing a result of
Tutte dealing with their proper q-colourings. Instatistical physics terms, we
solvethe q-state Potts model on random planar lattices. This work follows a
first paper by the same authors, where the generating functionwas proved to be
algebraic for certain values of q,including q=1, 2 and 3. It isknown to be
transcendental in general. In contrast, our differential system holds for an
indeterminate q.For certain special cases of combinatorial interest (four
colours; properq-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.Comment: 43 p
Bijective counting of Kreweras walks and loopless triangulations
We consider lattice walks in the plane starting at the origin, remaining in
the first quadrant and made of West, South and North-East steps. In 1965,
Germain Kreweras discovered a remarkably simple formula giving the number of
these walks (with prescribed length and endpoint). Kreweras' proof was very
involved and several alternative derivations have been proposed since then. But
the elegant simplicity of the counting formula remained unexplained. We give
the first purely combinatorial explanation of this formula. Our approach is
based on a bijection between Kreweras walks and triangulations with a
distinguished spanning tree. We obtain simultaneously a bijective way of
counting loopless triangulations.Comment: 25 page
Solution to a combinatorial puzzle arising from Mayer's theory of cluster integrals
Mayer's theory of cluster integrals allows one to write the partition
function of a gas model as a generating function of weighted graphs. Recently,
Labelle, Leroux and Ducharme have studied the graph weights arising from the
one-dimensional hard-core gas model and noticed that the sum of the weights
over all connected graphs with vertices is . This is, up to
sign, the number of rooted Cayley trees on vertices and the authors asked
for a combinatorial explanation. The main goal of this article is to provide
such an explanation.Comment: 9 page
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