Axel Bacher

Efficient random sampling of binary and unary-binary trees via holonomic equations

We present a new uniform random sampler for binary trees with

Weakly directed self-avoiding walks

n

The Dyck pattern poset

internal nodes consuming

Complexity of Anticipated Rejection Algorithms and the Darling---Mandelbrot Distribution

2n + \Theta(\log(n)^2)

Average site perimeter of directed animals on the two-dimensional lattices

random bits on average. This makes it quasi-optimal and out-performs the classical Remy algorithm. We also present a sampler for unary-binary trees with

Generating Random Permutations by Coin Tossing: Classical Algorithms, New Analysis, and Modern Implementation

n

Directed and multi-directed animals on the King’s lattice

nodes taking

Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Enumerative Aspects

\Theta(n)

Multivariate Lagrange inversion formula and the cycle lemma

random bits on average. Both are the first linear-time algorithms to be optimal up to a constant.

Exact-size sampling for Motzkin trees in linear time via Boltzmann samplers and holonomic specification

We define a new family of self-avoiding walks (SAW) on the square lattice, called weakly directed walks. These walks have a simple characterization in terms of the irreducible bridges that compose them. We determine their generating function. This series has a complex singularity structure and in particular, is not D-finite. The growth constant is approximately 2.54 and is thus larger than that of all natural families of SAW enumerated so far (but smaller than that of general SAW, which is about 2.64). We also prove that the end-to-end distance of weakly directed walks grows linearly. Finally, we study a diagonal variant of this model.