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

Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB-BA=1. This study surveys the relationships between classical combinatorial structures and the reduction to normal form of operator polynomials in such an algebra. The connection is achieved through suitable labelled graphs, or "diagrams", that are composed of elementary "gates". In this way, many normal form evaluations can be systematically obtained, thanks to models that involve set partitions, permutations, increasing trees, as well as weighted lattice paths. Extensions to q-analogues, multivariate frameworks, and urn models are also briefly discussed.

Abstract In this article we develop a vectorial kernel method—a powerful method which solves in a unified framework all the problems related to the enumeration of words generated by a pushdown automaton. We apply it for the enumeration of lattice paths that avoid a fixed word (a pattern), or for counting the occurrences of a given pattern. We unify results from numerous articles concerning patterns like peaks, valleys, humps, etc., in Dyck and Motzkin paths. This refines the study by Banderier and Flajolet from 2002 on enumeration and asymptotics of lattice paths: we extend here their results to pattern-avoiding walks/bridges/meanders/excursions. We show that the autocorrelation polynomial of this forbidden pattern, as introduced by Guibas and Odlyzko in 1981 in the context of rational languages, still plays a crucial role for our algebraic languages. En passant, our results give the enumeration of some classes of self-avoiding walks, and prove several conjectures from the On-Line Encyclopedia of Integer Sequences. Finally, we also give the trivariate generating function (length, final altitude, number of occurrences of the pattern p), and we prove that the number of occurrences is normally distributed and linear with respect to the length of the walk: this is what Flajolet and Sedgewick call an instance of Borges’s theorem.

We investigate the number of variables in two special subclasses of lambda-terms that are restricted by a bound of the number of abstractions between a variable and its binding lambda, and by a bound of the nesting levels of abstractions, respectively. These restrictions are on the one hand very natural from a practical point of view, and on the other hand they simplify the counting problem compared to that of unrestricted lambda-terms in such a way that the common methods of analytic combinatorics are applicable. We will show that the total number of variables is asymptotically normally distributed for both subclasses of lambda-terms with mean and variance asymptotically equal to C_1 n and C_2 n, respectively, where the constants C_1 and C_2 depend on the bound that has been imposed. So far we just derived closed formulas for the constants in case of the class of lambda-terms with a bounded number of abstractions between each variable and its binding lambda. However, for the other class of lambda-terms that we consider, namely lambda-terms with a bounded number of nesting levels of abstractions, we investigate the number of variables in the different abstraction levels and thereby exhibit very interesting results concerning the distribution of the variables within those lambda-terms.

John Tromp introduced the so-called 'binary lambda calculus' as a way to encode lambda terms in terms of binary words. Later, Grygiel and Lescanne conjectured that the number of binary lambda terms with $m$ free indices and of size $n$ (encoded as binary words of length $n$) is $o(n^{-3/2} τ^{-n})$ for $τ\approx 1.963448\ldots$. We generalize the proposed notion of size and show that for several classes of lambda terms, including binary lambda terms with $m$ free indices, the number of terms of size $n$ is $Θ(n^{-3/2} ρ^{-n})$ with some class dependent constant $ρ$, which in particular disproves the above mentioned conjecture. A way to obtain lower and upper bounds for the constant near the leading term is presented and numerical results for a few previously introduced classes of lambda terms are given.

Two stochastic processes describing the contour of simply generated random trees are studied: the contour process as defined by Gutjahr and Pflug [W. Gutjahr and G. Ch. Pflug, Stochastic Process. Appl., 41 (1992), pp. 69--89] and the traverse process constructed of the node heights during pre-order traversal of the tree. Using multivariate generating functions and singularity analysis the weak convergence of the contour process to Brownian excursion is shown and a new proof of the analogous result for the traverse process is obtained.