Related papers: Simply Generated Trees, B-series and Wigner Proces…
Any function can be constructed using a hierarchy of simpler functions through compositions. Such a hierarchy can be characterized by a binary rooted tree. Each node of this tree is associated with a function which takes as inputs two…
We determine the limit of the expected value and the variance of the protection number of the root in simply generated trees, in P\'olya trees, and in unlabelled non-plane binary trees, when the number of vertices tends to infinity.…
We introduce bud generating systems, which are used for combinatorial generation. They specify sets of various kinds of combinatorial objects, called languages. They can emulate context-free grammars, regular tree grammars, and synchronous…
We show that without other further assumption than affine equivariance and locality, a numerical integrator has an expansion in a generalized form of Butcher series (B-series) which we call aromatic B-series. We obtain an explicit…
We connect a primitive operation from arithmetic -- summing the digits of a base-$B$ integer -- to $q$-series and product generating functions analogous to those in partition theory. We find digit sum generating functions to be intertwined…
We study random unrooted plane trees with $n$ vertices sampled according to the weights corresponding to the vertex-degrees. Our main result shows that if the generating series of the weights has positive radius of convergence, then this…
We study self-similarity in random binary rooted trees. In a well-understood case of Galton-Watson trees, a distribution on a space of trees is said to be self-similar if it is invariant with respect to the operation of pruning, which cuts…
We obtain first order linear partial differential equations which are satisfied by exponential generating functions of two variables for the number of connected bipartite graphs with given Betti number. By solving these equations…
We consider various classes of Motzkin trees as well as lambda-terms for which we derive asymptotic enumeration results. These classes are defined through various restrictions concerning the unary nodes or abstractions, respectively: We…
A general class of stochastic Runge-Kutta methods for the weak approximation of It\^o and Stratonovich stochastic differential equations with a multi-dimensional Wiener process is introduced. Colored rooted trees are used to derive an…
We propose generating functions which encode the degeneracies and wall-crossing phenomena of $\mathcal{N}=2$ BPS structures. The generating functions have a representation-theoretic origin and are the analogs of the 1/4-BPS dyon counting…
Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often the generating tree for the set of permutations avoiding a set of patterns requires infinitely…
Relatively prime pairs of integers can be represented as nodes in three way branching trees. We construct trees of B\'ezout coefficients which correspond to the relatively prime pairs in the aforementioned trees. As one application, we…
This paper considers linear functions constructed on two different weighted branching processes and provides explicit bounds for their Kantorovich-Rubinstein distance in terms of couplings of their corresponding generic branching vectors.…
It is well-known that the height profile of a critical conditioned Galton-Watson tree with finite offspring variance converges, after a suitable normalization, to the local time of a standard Brownian excursion. In this work, we study the…
We discuss a method for computing the generating function for the multiplicity distribution in field theories with strong time dependent external sources. At leading order, the computation of the generating function reduces to finding a…
We propose a novel approach for studying rooted trees by using functions that we will call descent functions. We provide a construction method for rooted trees that allows to study their properties through the use of descent functions.…
We describe a generating tree approach to the enumeration and exhaustive generation of k-nonnesting set partitions and permutations. Unlike previous work in the literature using the connections of these objects to Young tableaux and…
The simple Galton--Watson process describes populations where individuals live one season and are then replaced by a random number of children. It can also be viewed as a way of generating random trees, each vertex being an individual of…
The enumeration of maps and the study of uniform random maps have been classical topics of combinatorics and statistical physics ever since the seminal work of Tutte in the sixties. Following the bijective approach initiated by Cori and…