Related papers: Chromatic statistics for Catalan and Fu{\ss}-Catal…
We will look at the Catalan numbers from the {\it Rigged Configurations} point of view originated \cite{Kir} from an combinatorial analysis of the Bethe Ansatz Equations associated with the higher spin anisotropic Heisenberg models . Our…
We provide some variations on the Greene-Krammer's identity which involve q-Catalan numbers. Our method reveals a curious analogy between these new identities and some congruences modulo a prime.
A "truncation" of Pascal's triangle is a triangular array of numbers that satisfies the usual Pascal recurrence but with a boundary condition that declares some terminal set of numbers along each row of the array to be zero. Presented here…
We give superexponential lower and upper bounds on the number of coloured $d$-dimensional triangulations whose underlying space is an oriented manifold, when the number of simplices goes to infinity and $d\geq 3$ is fixed. In the special…
The fractional calculus of variations and fractional optimal control are generalizations of the corresponding classical theories, that allow problem modeling and formulations with arbitrary order derivatives and integrals. Because of the…
We first give a geometric construction of a 2-dimensional mixed motive over $\mathbb{Q}$ with the Catalan constant $\mathbf{G}=1-1/3^2+1/5^2-1/7^2+\cdots$ as a period. We then use this motive to obtain a supply of linear forms in 1 and…
Given an ideal triangulation of a connected 3-manifold with non-empty boundary consisting of a disjoint union of tori, a point of the deformation variety is an assignment of complex numbers to the dihedral angles of the tetrahedra subject…
Around 10 years ago, Agol and Krushkal showed that the number of chromatic polynomials $P_{G}$ arising from graphs $G$ on $n$ vertices grows exponentially with $n$, by establishing that the (dual) flow polynomial…
A triangulation of a polygon is a subdivision of it into triangles, using diagonals between its vertices. Two different triangulations of a polygon can be related by a sequence of flips: a flip replaces a diagonal by the unique other…
In this paper, we extend the notion of labeled partitions with ordinary permutations to colored permutations in the sense that the colors are endowed with a cyclic structure. We use labeled partitions with colored permutations to derive the…
In this paper, we study metric completions of triangulated categories in a representation-theoretic context. We provide a concrete description of completions of bounded derived categories of hereditary finite dimensional algebras of finite…
Arslan, Altoum, and Zaarour introduced an inversion statistic for generalized symmetric groups. In this work, we study the distribution of this statistic over colored permutations, including derangements and involutions. By establishing a…
A Gaussian fluctuation formula is proved for linear statistics of complex random matrices in the case that the statistic is rotationally invariant. For a general linear statistic without this symmetry, Coulomb gas theory is used to predict…
In this paper we introduce a class of hypergraphs that we call chordal. We also extend the definition of triangulated hypergraphs, given in \cite{VT}, so that a triangulated hypergraph, according to our definition, is a natural…
We consider several generalizations of the classical $\gamma$-positivity of Eulerian polynomials (and their derangement analogues) using generating functions and combinatorial theory of continued fractions. For the symmetric group, we prove…
This paper studies increasing trees on $n$ labeled vertices, in which labels increase from the root to the leaves. It is known that the number of binary increasing trees coincides with the number of alternating permutations (Euler numbers).…
In this paper, we explain how the classical Catalan families of objects involving paths, tableaux, triangulations, parentheses configurations and more generalize canonically to a three-dimensional version. In particular, we present…
We prove the global triangulation conjecture for families of refined p-adic representations under a mild condition. That is, for a refined family, the associated family of (phi, Gamma)-modules admits a global triangulation on a Zariski open…
By means of a variational approach we find new series representations both for well known mathematical constants, such as $\pi$ and the Catalan constant, and for mathematical functions, such as the Riemann zeta function. The series that we…
A proper vertex colouring of a graph is \emph{nested} if the vertices of each of its colour classes can be ordered by inclusion of their open neighbourhoods. Through a relation to partially ordered sets, we show that the nested chromatic…