Related papers: Redundant generating functions in lattice path enu…
The main aim of this paper is to provide a novel approach to deriving identities for the Bernstein polynomials using functional equations. We derive various functional equations and differential equations using generating functions.…
We give two explicit sets of generators of the group of invertible regular functions over QQ on the modular curve Y1(N). The first set of generators is very surprising. It is essentially the set of defining equations of Y1(k) for k <= N/2…
We initiate the study of enumerating linear subspaces of alternating matrices over finite fields with explicit coordinates. We postulate that this study can be viewed as a linear algebraic analogue of the classical topic of enumerating…
In a previous paper [Adcock & Huybrechs, 2019] we described the numerical approximation of functions using redundant sets and frames. Redundancy in the function representation offers enormous flexibility compared to using a basis, but…
Latent representations are the essence of deep generative models and determine their usefulness and power. For latent representations to be useful as generative concept representations, their latent space must support latent space…
In this paper we study restricted overpartitions and concave compositions. In several cases the resulting generating functions involve simultaneously modular forms, mock theta functions, mock Maass theta functions, and false theta…
Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of…
In this paper, we construct a generating function quadratic at infinity for any exact Lagrangian in $\mathbb R^{2n}$ that equals $\mathbb R^n$ outside a compact set. Such a Lagrangian may be viewed as a Lagrangian filling of the standard…
Discovering latent representations of the observed world has become increasingly more relevant in data analysis. Much of the effort concentrates on building latent variables which can be used in prediction problems, such as classification…
Detection and elimination of redundant clauses from propositional formulas in Conjunctive Normal Form (CNF) is a fundamental problem with numerous application domains, including AI, and has been the subject of extensive research. Moreover,…
We define a random walk problem which admits analytic results, on a class of infinite periodic lattices which are directed and colored. Our approach is motivated from the fact that such lattices arise in string theoretic constructs of…
The biological roles of gene sets are used to group them into collections. These collections are often characterized by being high-dimensional, overlapping, and redundant families of sets, thus precluding a straightforward interpretation…
We investigate a functional equation which resembles the functional equation for the generating function of a lattice walk model for the quarter plane. The interesting feature of this equation is that its orbit sum is zero while its…
Most causal discovery algorithms find causal structure among a set of observed variables. Learning the causal structure among latent variables remains an important open problem, particularly when using high-dimensional data. In this paper,…
We give two combinatorial interpretations of the Matrix Ansatz of the PASEP in terms of lattice paths and rook placements. This gives two (mostly) combinatorial proofs of a new enumeration formula for the partition function of the PASEP.…
As countless examples show, it can be fruitful to study a sequence of complicated objects all at once via the formalism of generating functions. We apply this point of view to the homology and combinatorics of orbit configuration spaces:…
The aim of this paper is to study generating functions for the coefficients of the classical superoscillatory function associated with weak measurements. We also establish some new relations between the superoscillatory coefficients and…
By work of Bringmann, Ono, and Rhoades it is known that the generating function of the $M_2$-rank of partitions without repeated odd parts is the so-called holomorphic part of a certain harmonic Maass form. Here we improve the standing of…
Mullineux defined an involution on the set of $e$-regular partitions of $n$. When $e=p$ is prime, these partitions label irreducible symmetric group modules in characteristic $p$. Mullineux's conjecture, since proven, was that this…
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…