183 citations · 362 across the 15 of their papers we have counts for
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Enumeration of three-quadrant walks via invariants: some diagonally symmetric models
Mireille Bousquet-Mélou
In the past 20 years, the enumeration of plane lattice walks confined to a convex cone -- normalized into the first quadrant -- has received a lot of attention, stimulated the deve…
Walks avoiding a quadrant and the reflection principle
Mireille Bousquet-Mélou, Michael Wallner
We continue the enumeration of plane lattice walks with small steps avoiding the negative quadrant, initiated by the first author in 2016. We solve in detail a new case, namely the…
Plane bipolar orientations and quadrant walks
Mireille Bousquet-Mélou, Éric Fusy, Kilian Raschel
Bipolar orientations of planar maps have recently attracted some interest in combinatorics, probability theory and theoretical physics. Plane bipolar orientations with edges ar…
Eulerian orientations and the six-vertex model on planar map
Mireille Bousquet-Mélou, Andrew Elvey Price, Paul Zinn-Justin
We address the enumeration of planar 4-valent maps equipped with an Eulerian orientation by two different methods, and compare the solutions we thus obtain. With the first method w…
Counting walks with large steps in an orthant
Alin Bostan, Mireille Bousquet-Mélou, Stephen Melczer
In the past fifteen years, the enumeration of lattice walks with steps takenin a prescribed set S and confined to a given cone, especially the firstquadrant of the plane, has been…
The generating function of planar Eulerian orientations
Mireille Bousquet-Mélou, Andrew Elvey Price
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 4-valent…
An elementary solution of Gessel's walks in the quadrant
Mireille Bousquet-Mélou
Around 2000, Ira Gessel conjectured that the number of lattice walks in the quadrant N^2, starting and ending at the origin (0,0) and taking their steps in {E,NE,W,SW} had a simple…
The representation of the symmetric group on m-Tamari intervals
Mireille Bousquet-Mélou, Guillaume Chapuy, Louis-François Préville-Ratelle
An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my}. The set…
Tamari lattices and parking functions: proof of a conjecture of F. Bergeron
Mireille Bousquet-Mélou, Guillaume Chapuy, Louis-François Préville Ratelle
An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my. The set…
Exactly solved models of polyominoes and polygons
Mireille Bousquet-Mélou, Richard Brak
This chapter deals with the exact enumeration of certain classes of self-avoiding polygons and polyominoes on the square lattice. We present three general approaches that apply to…
Walks with small steps in the quarter plane
Mireille Bousquet-Mélou, Marni Mishna
Let S be a subset of {-1,0,1}^2 not containing (0,0). We address the enumeration of plane lattice walks with steps in S, that start from (0,0) and always remain in the first quadra…
Rational and algebraic series in combinatorial enumeration
Mireille Bousquet-Mélou
Let A be a class of objects, equipped with an integer size such that for all n the number a(n) of objects of size n is finite. We are interested in the case where the generating fu…
Discrete excursions
Mireille Bousquet-Mélou
It is well-known that the length generating function E(t) of Dyck paths (excursions with steps +1 and -1) satisfies 1-E+t^2E^2=0. The generating function E^(k)(t) of Dyck paths of…
Three osculating walkers
Mireille Bousquet-Mélou
We consider three directed walkers on the square lattice, which move simultaneously at each tick of a clock and never cross. Their trajectories form a non-crossing configuration of…
Walks in the quarter plane: Kreweras' algebraic model
Mireille Bousquet-Mélou
We consider planar lattice walks that start from (0,0), remain inthe first quadrant i, j >= 0, and are made of three types of steps: North-East, West and South. These walks are kno…