Related papers: An algorithmic approach based on generating trees …
Trees are useful entities allowing to model data structures and hierarchical relationships in networked decision systems ubiquitously. An ordered tree is a rooted tree where the order of the subtrees (children) of a node is significant. In…
We consider the enumeration of pattern-avoiding involutions, focusing in particular on sets defined by avoiding a single pattern of length 4. As we demonstrate, the numerical data for these problems demonstrates some surprising behavior.…
A Skolem sequence is a linear arrangement of the multiset, {1, 1, 2, 2, ..., n, n} such that if r in [n] appears in positions i and j, then |i-j| = r. We first translate the problem to a particular set of perfect matchings, then apply the…
The structure of an evolving network contains information about its past. Extracting this information efficiently, however, is, in general, a difficult challenge. We formulate a fast and efficient method to estimate the most likely history…
We focus on generative AI for a type of data that still represent one of the most prevalent form of data: tabular data. Our paper introduces two key contributions: a new powerful class of forest-based models fit for such tasks and a simple…
Working with generating functions, the combinatorics of a recurrence relation can be expressed in a way that allows for more efficient calculation of the quantity. This is true of the Catalan numbers for an ordered binary tree…
We study joint distributions of cycles and patterns in permutations written in standard cycle form. We explore both classical and generalised patterns of length 2 and 3. Many extensions of classical theory are achieved; bivariate generating…
Despite the fact that the field of pattern avoiding permutations has been skyrocketing over the last two decades, there are very few exhaustive generating algorithms for such classes of permutations. In this paper we introduce the notions…
We consider the well-studied pattern counting problem: given a permutation $\pi \in \mathbb{S}_n$ and an integer $k > 1$, count the number of order-isomorphic occurrences of every pattern $\tau \in \mathbb{S}_k$ in $\pi$. Our first result…
Using the approach suggested in [arXiv:1002.2761] we present below a sufficient condition guaranteeing that two collections of patterns of permutations have the same exponential generating functions for the number of permutations avoiding…
Vincular and covincular patterns are generalizations of classical patterns allowing restrictions on the indices and values of the occurrences in a permutation. In this paper we study the integer sequences arising as the enumerations of…
Ascent sequences are sequences of nonnegative integers with restrictions on the size of each letter, depending on the number of ascents preceding it in the sequence. Ascent sequences have recently been related to (2+2)-free posets and…
We present an algorithm for learning decision trees using stochastic gradient information as the source of supervision. In contrast to previous approaches to gradient-based tree learning, our method operates in the incremental learning…
Labeled infinite trees provide combinatorial interpretations for many integer sequences generated by nested recurrence relations. Typically, such sequences are monotone increasing. Several of these sequences also have straightforward…
We propose methods for density estimation and data synthesis using a novel form of unsupervised random forests. Inspired by generative adversarial networks, we implement a recursive procedure in which trees gradually learn structural…
We prove a new formula for the generating function of multitype Cayley trees counted according to their degree distribution. Using this formula we recover and extend several enumerative results about trees. In particular, we extend some…
Decision trees are widely used for non-linear modeling, as they capture interactions between predictors while producing inherently interpretable models. Despite their popularity, performing inference on the non-linear fit remains largely…
Motivated by a correlation between the distribution of descents over permutations that avoid a consecutive pattern and those avoiding the respective quasi-consecutive pattern, as established in this paper, we obtain a complete $\des$-Wilf…
A new tree model is introduced based on ordered trees, by distinguishing exactly one child of each node that \emph{has} children. The basic enumeration leads to a cubic equation of the generating function. The extraction of its coefficients…
In this paper, we study the Wilf-type equivalence relations among multiset permutations. We identify all multiset equivalences among pairs of patterns consisting of a pattern of length three and another pattern of length at most four. To…