Related papers: Two deformed Pascal's triangles and its new proper…
We point out the joint occurrence of Pascal triangle patterns and power-law scaling in the standard logistic map, or more generally, in unimodal maps. It is known that these features are present in its two types of bifurcation cascades:…
In two previous papers we have presented partition formulae for the Fibonacci numbers motivated by the appearance of the Fibonacci numbers in the representation theory of the 3-Kronecker quiver and its universal cover, the 3-regular tree.…
We use Jones-Wenzl idempotents to construct bases for the relative Kauffman bracket skein module of a square with n points colored 1 and one point colored h. We consider a natural bilinear form on this skein module. We calculate the…
We study two types of regularizations of the determinant of Laplacian on Riemann manifold from the viewpoint of resurgence theory. One is the formal logarithmic derivative of the determinant, and the other is its exponential deformation.…
A new family of asymmetric matrices of Walsh-Hadamard type is introduced. We study their properties and, in particular, compute their determinants and discuss their eigenvalues. The invertibility of these matrices implies that certain…
In this paper we develop a new geometric method to answer the log-concavity questions related to a nice class of combinatorial sequences arising from the Pascal triangle.
Given a real cubic form f(x,y,z), there is a pseudo-Riemannian metric given by its Hessian matrix, defined on the open subset of R^3 where the Hessian determinant h is non-zero. We determine the full curvature tensor of this metric in terms…
Motivated by representation theory we exhibit an interior structure to Catalan sequences and many generalisations thereof. Certain of these coincide with well known (but heretofore isolated) structures. The remainder are new.
We study combinatorial properties of a rational Dyck path by decomposing it into a tuple of Dyck paths. The combinatorial models such as $b$-Stirling permutations, $(b+1)$-ary trees, parenthesis presentations, and binary trees play central…
The Catalan triangle, as well as a Fuss-Catalan triangle, enter a problem of counting particular tied arc diagrams. This setting allows us to prove some combinatorial properties of these triangles.
Catalan numbers and their interpretations in terms of Dyck paths are widely used in different topics of applied mathematics and computer science. Here, we consider a general approach for constrained Dyck paths. In particular, we study Dyck…
We consider four examples of combinatorial triangles $\left(T(n,k)\right)_{0\le k\le n}$ (Pascal, Stirling of both types, Euler) : through saddle-point asymptotics, their \emph{Pascal's formulas} define four vector fields, together with…
Smith normal form evaluations found by Bessenrodt and Stanley for some Hankel matrices of q-Catalan numbers are proven in two ways. One argument generalizes the Bessenrodt-Stanley results for the Smith normal form of a certain multivariate…
In this paper we establish plenty of number theoretic and combinatoric identities involving generalized Bernoulli and Stirling numbers of both kinds. These formulas are deduced from Pascal type matrix representations of Bernoulli and…
We study determinants of matrices whose entries are powers of Fibonacci numbers. We then extend the results to include entries that are powers of generalized Fibonacci numbers defined as a second-order linear recurrence relation. These…
In 1640's, Blaise Pascal discovered a remarkable property of a hexagon inscribed in a conic - Pascal Theorem, which gave birth of the projective geometry. In this paper, a new geometric invariant of algebraic curves is discovered by a…
In this paper, we introduce two new forms of the dual Hartwig-Spindelb{\"o}ck decomposition and employ them to derive explicit representations for several classes of dual generalized inverses. Building on these representations, we further…
A Hadamard-Hitchcock decomposition of a multidimensional array is a decomposition that expresses the latter as a Hadamard product of several tensor rank decompositions. Such decompositions can encode probability distributions that arise…
The Baker-Campbell-Hausdorff formula was recently resummed exactly in one variable, and left as a power series in the other (Moodie and Long 2021 J. Phys. A: Math. Theor. 54 015208). The coefficients of the power series were provided as a…
We introduce deformations of Kazhdan-Lusztig elements and specialised nonsymmetric Macdonald polynomials, both of which form a distinguished basis of the polynomial representation of a maximal parabolic subalgebra of the Hecke algebra. We…