Olivier Bernardi

Intervals in Catalan lattices and realizers of triangulations

The Stanley lattice, Tamari lattice and Kreweras lattice are three remarkable orders defined on the set of Catalan objects of a given size. These lattices are ordered by inclusion: the Stanley lattice is an extension of the Tamari lattice which is an extension of the Kreweras lattice. The Stanley order can be defined on the set of Dyck paths of size n as the relation of being above. Hence, intervals in the Stanley lattice are pairs of non-crossing Dyck paths. In a previous article, the second author defined a bijection @F between pairs of non-crossing Dyck paths and the realizers of triangulations (or Schnyder woods). We give a simpler description of the bijection @F. Then, we study the restriction of @F to Tamari and Kreweras intervals. We prove that @F induces a bijection between Tamari intervals and minimal realizers. This gives a bijection between Tamari intervals and triangulations. We also prove that @F induces a bijection between Kreweras intervals and the (unique) realizers of stack triangulations. Thus, @F induces a bijection between Kreweras intervals and stack triangulations which are known to be in bijection with ternary trees.

A bijection for covered maps, or a shortcut between Harer-Zagier's and Jackson's formulas

We consider maps on orientable surfaces. A map is called unicellular if it has a single face. A covered map is a map (of genus g) with a marked unicellular spanning submap (which can have any genus in {0,1,...,g}). Our main result is a bijection between covered maps with n edges and genus g and pairs made of a plane tree with n edges and a unicellular bipartite map of genus g with n+1 edges. In the planar case, covered maps are maps with a marked spanning tree and our bijection specializes into a construction obtained by the first author in Bernardi (2007) [4]. Covered maps can also be seen as shuffles of two unicellular maps (one representing the unicellular submap, the other representing the dual unicellular submap). Thus, our bijection gives a correspondence between shuffles of unicellular maps, and pairs made of a plane tree and a unicellular bipartite map. In terms of counting, this establishes the equivalence between a formula due to Harer and Zagier for general unicellular maps, and a formula due to Jackson for bipartite unicellular maps. We also show that the bijection of Bouttier, Di Francesco and Guitter (2004) [8] (which generalizes a previous bijection by Schaeffer, 1998 [33]) between bipartite maps and so-called well-labeled mobiles can be obtained as a special case of our bijection.

Schnyder Decompositions for Regular Plane Graphs and Application to Drawing

Schnyder woods are decompositions of simple triangulations into three edge-disjoint spanning trees crossing each other in a specific way. In this article, we generalize the definition of Schnyder woods to d-angulations (plane graphs with faces of degree d) for all d≥3. A Schnyder decomposition is a set of d spanning forests crossing each other in a specific way, and such that each internal edge is part of exactly d−2 of the spanning forests. We show that a Schnyder decomposition exists if and only if the girth of the d-angulation is d. As in the case of Schnyder woods (d=3), there are alternative formulations in terms of orientations (“fractional” orientations when d≥5) and in terms of corner-labellings. Moreover, the set of Schnyder decompositions of a fixed d-angulation of girth d has a natural structure of distributive lattice. We also study the dual of Schnyder decompositions which are defined on d-regular plane graphs of mincut d with a distinguished vertex v∗: these are sets of d spanning trees rooted at v∗ crossing each other in a specific way and such that each edge not incident to v∗ is used by two trees in opposite directions. Additionally, for even values of d, we show that a subclass of Schnyder decompositions, which are called even, enjoy additional properties that yield a reduced formulation; in the case d=4, these correspond to well-studied structures on simple quadrangulations (2-orientations and partitions into 2 spanning trees).In the case d=4, we obtain straight-line and orthogonal planar drawing algorithms by using the dual of even Schnyder decompositions. For a 4-regular plane graph G of mincut 4 with a distinguished vertex v∗ and n−1 other vertices, our algorithms places the vertices of G\v∗ on a (n−2)×(n−2) grid according to a permutation pattern, and in the orthogonal drawing each of the 2n−4 edges of G\v∗ has exactly one bend. The vertex v∗ can be embedded at the cost of 3 additional rows and columns, and 8 additional bends.We also describe a further compaction step for the drawing algorithms and show that the obtained grid-size is strongly concentrated around 25n/32×25n/32 for a uniformly random instance with n vertices.

Counting trees using symmetries

We prove a new formula for the generating function of multitype Cayley trees counted according to their degree distribution. Using this formula we recover and extend several enumerative results about trees. In particular, we extend some results by Knuth and by Bousquet-Melou and Chapuy about embedded trees. We also give a new proof of the multivariate Lagrange inversion formula. Our strategy for counting trees is to exploit symmetries of refined enumeration formulas: proving these symmetries is easy, and once the symmetries are proved the formulas follow effortlessly. We also adapt this strategy to recover an enumeration formula of Goulden and Jackson for cacti counted according to their degree distribution.

Enumerating simplicial decompositions of surfaces with boundaries

It is well-known that the triangulations of the disc with n+2 vertices on its boundary are counted by the nth Catalan number C(n)=1n+12nn. This paper deals with the generalisation of this problem to any compact surface S with boundaries. We obtain the asymptotic number of simplicial decompositions of the surface S with n vertices on its boundary. More generally, we determine the asymptotic number of dissections of S when the faces are @d-gons with @d belonging to a set of admissible degrees @D@?{3,4,5,...}. We also give the limit laws for certain parameters of such dissections.

A Short Proof of Rayleigh's Theorem with Extensions

Abstract Consider a walk in the plane made of n unit steps, with directions chosen independently and uniformly at random at each step. Rayleighs theorem asserts that the probability for such a walk to end at a distance less than 1 from its starting point is 1/(n + 1). We give an elementary proof of this result. We also prove the following generalization, valid for any probability distribution μ on the positive real numbers: If two walkers start at the same point and make, respectively, m and n independent steps with uniformly random directions and with lengths chosen according to μ, then the probability that the first walker ends farther away than the second is m/(m + n).

Counting coloured planar maps: differential equations

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

A bijection for covered maps on orientable surfaces

{q=1, 2}