arXiv subjects

Axel Bacher

Publications and source records attributed to Axel Bacher.

12 recordsLinked to original sources

arXiv2026

Families of Eliahou semigroups linked to Farey intervals

We describe a way to explore the numerical semigroup tree to find all Eliahou semigroups of conductor up to 200 (and conjecturally up to 400) thanks to a new way of representing the semigroups and pruning off unwanted branches. This proves that Wilf's conjecture holds for conductor up to 200. Based on this list, we describe new classes of numerical semigroups, with a Farey interval as a crucial parameter. We show that these semigroups probably all satisfy Wilf's conjecture. We explicitly construct many families of Eliahou semigroups belonging to these classes, encompassing families described by Delgado, Eliahou and Fromentin, and Bras-Amor\'os.

math.CO↗

arXiv2024

Progressive and Rushed Dyck Paths

We call progressive paths and rushed paths two families of Dyck paths studied by Asinowski and Jelinek, which have the same enumerating sequence (OEIS entry A287709). We present a bijection proving this fact. Rushed paths turn out to be in bijection with one-sided trees, introduced by Durhuus and Unel, which have an asymptotic enumeration involving a stretched exponential. We conclude by presenting several other classes of related lattice paths and directed animals that may have similar asymptotic properties.

math.CO↗

arXiv2018

Improving the Florentine algorithms: recovering algorithms for Motzkin and Schröder paths

We present random sampling procedures for Motzkin and Schröder paths, following previous work on Dyck paths. Our algorithms follow the anticipated rejection method of the Florentine algorithms (Barcucci et al. 1994+), but introduce a recovery idea to greatly reduce the probability of rejection. They use an optimal amount of randomness and achieve a better time complexity than the Florentine algorithms.

cs.DS↗

arXiv2016

A new bijection on m-Dyck paths with application to random sampling

We present a new bijection between variants of $m$-Dyck paths (paths with steps in $\{+1,-m\}$ starting and ending at height $0$ and remaining at non-negative height), which generalizes a classical bijection between Dyck prefixes and pointed Łukasiewicz paths. As an application, we present a new random sampling procedure for $m$-Dyck paths with a linear time complexity and using a quasi-optimal number of random bits. This outperforms Devroye's algorithm, which uses $\mathcal O(n\log n)$ random bits.

math.CO↗

arXiv2015

Continued Classification of 3D Lattice Walks in the Positive Octant

We continue the investigations of lattice walks in the three dimensional lattice restricted to the positive octant. We separate models which clearly have a D-finite generating function from models for which there is no reason to expect that their generating function is D-finite, and we isolate a small set of models whose nature remains unclear and requires further investigation. For these, we give some experimental results about their asymptotic behaviour, based on the inspection of a large number of initial terms. At least for some of them, the guessed asymptotic form seems to tip the balance towards non-D-finiteness.

math.CO↗

arXiv2015

Directed and multi-directed animals on the king's lattice

This article introduces a new, simple solvable lattice for directed animals: the directed king's lattice, or square lattice with next nearest neighbor bonds and preferred directions {W, NW, N, NE, E}. We show that the directed animals in this lattice have an algebraic generating function linked to the Schröder numbers and belong to the same universality class as the ones in the square and triangular lattices. We also define multi-directed animals in the king's lattice, which form a superclass of directed animals. We compute their generating function and show that it is not D-finite. Finally, we propose efficient random sampling algorithms for our animals.

math.CO↗

arXiv2015

Complexity of Anticipated Rejection Algorithms and the Darling-Mandelbrot Distribution

We study in limit law the complexity of some anticipated rejection random sampling algorithms. We express this complexity in terms of a probabilistic process, the threshold sum process. We show that, under the right conditions, the complexity is linear and admits as a limit law a so-called Darling-Mandelbrot distribution, studied by Darling (Trans Am Math Soc 73:95-107, 1952) and Lew (Constr Approx 10(1):15-30, 1994). We also give an explicit form to the density of the Darling-Mandelbrot distribution and derive some of its analytic properties.

math.CO↗

arXiv2015

MergeShuffle: A Very Fast, Parallel Random Permutation Algorithm

This article introduces an algorithm, MergeShuffle, which is an extremely efficient algorithm to generate random permutations (or to randomly permute an existing array). It is easy to implement, runs in $n\log_2 n + O(1)$ time, is in-place, uses $n\log_2 n + Θ(n)$ random bits, and can be parallelized accross any number of processes, in a shared-memory PRAM model. Finally, our preliminary simulations using OpenMP suggest it is more efficient than the Rao-Sandelius algorithm, one of the fastest existing random permutation algorithms. We also show how it is possible to further reduce the number of random bits consumed, by introducing a second algorithm BalancedShuffle, a variant of the Rao-Sandelius algorithm which is more conservative in the way it recursively partitions arrays to be shuffled. While this algorithm is of lesser practical interest, we believe it may be of theoretical value. Our full code is available at: https://github.com/axel-bacher/mergeshuffle

cs.DS↗

arXiv2014

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

We present a new uniform random sampler for binary trees with $n$ internal nodes consuming $2n + Θ(\log(n)^2)$ 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 $n$ nodes taking $Θ(n)$ random bits on average. Both are the first linear-time algorithms to be optimal up to a constant.

cs.DS↗

arXiv2013

Generalized Dyck paths of bounded height

Generalized Dyck paths (or discrete excursions) are one-dimensional paths that take their steps in a given finite set S, start and end at height 0, and remain at a non-negative height. Bousquet-Mélou showed that the generating function E_k of excursions of height at most k is of the form F_k/F_{k+1}, where the F_k are polynomials satisfying a linear recurrence relation. We give a combinatorial interpretation of the polynomials F_k and of their recurrence relation using a transfer matrix method. We then extend our method to enumerate discrete meanders (or paths that start at 0 and remain at a non-negative height, but may end anywhere). Finally, we study the particular case where the set S is symmetric and show that several simplifications occur.

math.CO↗

arXiv2011

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

We introduce new combinatorial (bijective) methods that enable us to compute the average value of three parameters of directed animals of a given area, including the site perimeter. Our results cover directed animals of any one-line source on the square lattice and its bounded variants, and we give counterparts for most of them in the triangular lattices. We thus prove conjectures by Conway and Le Borgne. The techniques used are based on Viennot's correspondence between directed animals and heaps of pieces (or elements of a partially commutative monoid).

math.CO↗

arXiv2010

Weakly directed self-avoiding walks

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.

math.CO↗