Related papers: Dyck tilings of type $D$
K-theoretic Donaldson invariants are holomorphic Euler characteristics of determinant line bundles on moduli spaces of sheaves on surfaces. We compute generating functions of K-theoretic Donaldson invariants on the projective plane and…
We obtain structural results on translational tilings of periodic functions in $\mathbb{Z}^d$ by finite tiles. In particular, we show that any level one tiling of a periodic set in $\mathbb{Z}^2$ must be weakly periodic (the disjoint union…
The purpose of this paper is to present a collection of interesting generating functions for partitions which have connections to positive definite binary quadratic forms. In establishing our results we obtain some new Bailey pairs.
In this thesis, we introduce the subject of D-spaces and some of its most important open problems which are related to well known covering properties. We then introduce a new approach for studying D-spaces and covering properties in…
We define a generalized vector partition function and derive an identity for generating series of such functions associated with solutions of basic recurrence relation of combinatorial analysis. As a consequence, we obtain the generating…
A new method for constructing aperiodic tilings is presented. The method is illustrated by constructing a particular tiling and its hull. The properties of this tiling and the hull are studied. In particular it is shown that these tilings…
We present the generating function for the numbers of isomorphism classes of coverings of the two-dimensional sphere by the genus $g$ compact oriented surface not ramified outside of a given set of $m+1$ points in the target, fixed…
We introduce type C parking functions, encoded as vertically labelled lattice paths and endowed with a statistic dinv'. We define a bijection from type C parking functions to regions of the Shi arrangement of type C, encoded as diagonally…
We address the enumeration of q-coloured planar maps counted bythe number of edges and the number of monochromatic edges. We prove that the associated generating function is differentially algebraic,that is, satisfies a non-trivial…
A generating function for the Wigner's $D$-matrix elements of $SU(3)$ is derived. From this an explicit expression for the individual matrix elements is obtained in a closed form.
Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the…
Let $m,k$ be fixed positive integers. Determining the generating function for the number of tilings of an $m\times n$ rectangle by $k\times 1$ rectangles is a long-standing open problem to which the answer is only known in certain special…
Two methods of constructing 2D Toda $\tau$-functions that are generating functions for certain geometrical invariants of a combinatorial nature are related. The first involves generation of paths in the Cayley graph of the symmetric group…
Our earlier article proved that if $n > 1$ translates of sublattices of $Z^d$ tile $Z^d$, and all the sublattices are Cartesian products of arithmetic progressions, then two of the tiles must be translates of each other. We re-prove this…
We introduce and prove some basic results about a combinatorial model which produces embedded polygons in the plane. The model is closely related to outer billiards on kites, and also is related to corner percolation, to Hooper's Truchet…
We study tilting complexes over preprojective algebras of Dynkin type. We classify all tilting complexes by giving a bijection between tilting complexes and the braid group of the corresponding folded graph. In particular, we determine the…
The $d$-fold ($d \geq 3$) branched coverings on a disk give an infinite family of nongeometric embeddings of braid groups into mapping class groups. We, in this paper, give new explicit expressions of these braid group representations into…
We present in this paper a new family of implicit function for synthesizing a wide variety of 3D surfaces. The basis of this family consists of the usual functions that are: the function rectangular pulses, the function saw-tooth pulses,…
Much of the human cortex seems to be organized into topographic cortical maps. Yet few quantitative methods exist for characterizing these maps. To address this issue we developed a modeling framework that can reveal group-level cortical…
We consider the problem of enumerating d-irreducible maps, i.e. planar maps whose all cycles have length at least d, and such that any cycle of length d is the boundary of a face of degree d. We develop two approaches in parallel: the…