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The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with $m$ flaws is the $n$-th Catalan number and independent on $m$. In this paper, we consider the refinements of Dyck paths with flaws by four…

We consider a class of multiplicative processes which, added with stochastic reset events, give origin to stationary distributions with power-law tails -- ubiquitous in the statistics of social, economic, and ecological systems. Our main…

A number system coding for the permutations generated by cyclic shift is described. The system allows to find the rank of a permutation given how it has been generated, and to determine a permutation given its rank. It defines a code…

Over the last two decades, network theory has shown to be a fruitful paradigm in understanding the organization and functioning of real-world complex systems. One technique helpful to this endeavor is identifying functionally influential…

In this paper we outline a Matrix Ansatz approach to some problems of combinatorial enumeration. The idea is that many interesting quantities can be expressed in terms of products of matrices, where the matrices obey certain relations. We…

We introduce a class of generative network models that insert edges by connecting the starting and terminal vertices of a random walk on the network graph. Within the taxonomy of statistical network models, this class is distinguished by…

Parametric prediction error methods constitute a classical approach to the identification of linear dynamic systems with excellent large-sample properties. A more recent regularized approach, inspired by machine learning and Bayesian…

A hybrid model for biomedical time series comprising a continuous second-order linear time-invariant system driven by an input sequence of positively weighted Dirac delta-functions is considered. The problem of the joint estimation of the…

Catalan numbers arise in many enumerative contexts as the counting sequence of combinatorial structures. In this work, we consider natural Markov chains on some of the realizations of the Catalan sequence. While our main result is in…

We introduce the concept of a \emph{cycle pattern} for directed graphs as functions from the set of cycles to the set $\{-,0,+\}$. The key example for such a pattern is derived from a weight function, giving rise to the sign of the total…

We consider weighted small step walks in the positive quadrant, and provide algebraicity and differential transcendence results for the underlying generating functions: we prove that depending on the probabilities of allowed steps, certain…

We introduce the concept of pattern graphs--directed acyclic graphs representing how response patterns are associated. A pattern graph represents an identifying restriction that is nonparametrically identified/saturated and is often a…

Axiomatizing covarieties of coalgebras for an endofunctor is less intuitive than axiomatizing varieties of algebras via equations (Dahlqvist and Schmid, 2022). Existing techniques come from coalgebraic modal logic, pattern avoidance…

Weighted ensemble (WE) is an enhanced path-sampling method that is conceptually simple, widely applicable, and statistically exact. In a WE simulation, an ensemble of trajectories is periodically pruned or replicated to enhance sampling of…

In this paper, we present an overview of constrained PARAFAC models where the constraints model linear dependencies among columns of the factor matrices of the tensor decomposition, or alternatively, the pattern of interactions between…

One of the major successes in computational biology has been the unification, using the graphical model formalism, of a multitude of algorithms for annotating and comparing biological sequences. Graphical models that have been applied…

We find a bijection between bi-banded paths and peak-counting paths, applying to two classes of lattice paths including Dyck paths. Thus we find a new interpretation of Narayana numbers as coefficients of weight polynomials enumerating…

We discuss the complexity of path enumeration and counting in weighted temporal graphs. In a weighted temporal graph, each edge has an availability time, a traversal time and some real cost. We introduce two bicriteria temporal min-cost…

We consider the problem of counting the set of $\mathscr{D}_{a,b}$ of Dyck paths inscribed in a rectangle of size $a\times b$. They are a natural generalization of the classical Dyck words enumerated by the Catalan numbers. By using Ferrers…

Dynamical processes can be classified in various ways as deterministic or stochastic, and continuous or discrete time. All these types can be studied by the path-spaces they generate, and stationary measures on that path-space. Such…