Related papers: Computational complexity of counting coincidences
We consider the problem of enumeration of planar maps and revisit its one-matrix model solution in the light of recent combinatorial techniques involving conjugated trees. We adapt and generalize these techniques so as to give an…
Many natural counting problems arise in connection with the normal form of braids--and seem to have never been considered so far. Here we solve some of them by analysing the normality condition in terms of the associated permutations, their…
We present a detailed study of the combinatorial interpretation of matrix integrals, including the examples of tessellations of arbitrary genera, and loop models on random surfaces. After reviewing their methods of solution, we apply these…
We consider colored compositions where only some parts are allowed different colors, depending on their locations in the composition. The counting sequences are obtained through generating functions. Connections to many other combinatorial…
We prove a complexity dichotomy for a class of counting problems expressible as bipartite 3-regular Holant problems. For every problem of the form $\operatorname{Holant}\left(f\mid =_3 \right)$, where $f$ is any integer-valued ternary…
Can a probabilistic gambler get arbitrarily rich when all deterministic gamblers fail? We study this problem in the context of algorithmic randomness, introducing a new notion -- almost everywhere computable randomness. A binary sequence…
In many everyday categories (sets, spaces, modules, ...) objects can be both added and multiplied. The arithmetic of such objects is a challenge because there is usually no subtraction. We prove a family of cases of the following principle:…
Kasteleyn counted the number of domino tilings of a rectangle by considering a mutation of the adjacency matrix: a Kasteleyn matrix K. In this paper we present a generalization of Kasteleyn matrices and a combinatorial interpretation for…
The computation of Kronecker coefficients is a challenging problem with a variety of applications. In this paper we present an approach based on methods from symplectic geometry and residue calculus. We outline a general algorithm for the…
Whether or not the Kronecker coefficients of the symmetric group count some set of combinatorial objects is a longstanding open question. In this work we show that a given Kronecker coefficient is proportional to the rank of a projector…
We propose a new family of combinatorial inference problems for graphical models. Unlike classical statistical inference where the main interest is point estimation or parameter testing, combinatorial inference aims at testing the global…
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…
We revisit the problem of enumeration of vertex-tricolored planar random triangulations solved in [Nucl. Phys. B 516 [FS] (1998) 543-587] in the light of recent combinatorial developments relating classical planar graph counting problems to…
Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry…
Framed combinatorial topology is a novel theory describing combinatorial phenomena arising at the intersection of stratified topology, singularity theory, and higher algebra. The theory synthesizes elements of classical combinatorial…
The invariant polytope algorithm was a breakthrough in the joint spectral radius computation, allowing to find the exact value of the joint spectral radius for most matrix families~\cite{GP2013,GP2016}. This algorithm found many…
In these lecture notes we present some connections between random matrices, the asymmetric exclusion process, random tilings. These three apparently unrelated objects have (sometimes) a similar mathematical structure, an interlacing…
An ordered $r$-matching is an $r$-uniform hypergraph matching equipped with an ordering on its vertices. These objects can be viewed as natural generalisations of $r$-dimensional orders. The theory of ordered 2-matchings is well-developed…
A binary matrix satisfies the consecutive ones property (COP) if its columns can be permuted such that the ones in each row of the resulting matrix are consecutive. Equivalently, a family of sets F = {Q_1,..,Q_m}, where Q_i is subset of R…
A fixed set of vertices in the plane may have multiple planar straight-line triangulations in which the degree of each vertex is the same. As such, the degree information does not completely determine the triangulation. We show that even if…