Related papers: On enumeration of $l$-hypermaps
Recently, by the Riordan's identity related to tree enumerations, \begin{eqnarray*} \sum_{k=0}^{n}\binom{n}{k}(k+1)!(n+1)^{n-k} &=& (n+1)^{n+1}, \end{eqnarray*} Sun and Xu derived another analogous one, \begin{eqnarray*}…
We consider a construction of C*-algebras from continuous piecewise monotone maps on the circle which generalizes the crossed product construction for homeomorphisms and more generally the construction of Renault, Deaconu and…
Let $k$, $t$ and $m$ be positive integers. A $k$-multiset of $[m]$ is a collection of $k$ integers from the set $\{1,...,m\}$ in which the integers can appear more than once. We use graph homomorphisms and existing theorems for intersecting…
We prove Levinson's theorem for scattering on an (m+n)-vertex graph with n semi-infinite paths each attached to a different vertex, generalizing a previous result for the case n=1. This theorem counts the number of bound states in terms of…
Beginning from the resolution of Dirichlet L function, using the inner product formula of infinite-dimensional vectors in the complex space, the author proved the world's baffling problem--Generalized Riemann hypothesis.
Given a family of hypergraphs $\mathcal{H}$, we say that a hypergraph $\Gamma$ is $\mathcal{H}$-universal if it contains every $H \in \mathcal{H}$ as a subgraph. For $D, r \in \mathbb{N}$, we construct an $r$-uniform hypergraph with…
We present two hypermatrix formulations of the Cayley Hamilton theorem. One of the proposed formulation naturally extends to hypermatrices the combinatorial interpretations of the classical Cayley Hamilton theorem. We conclude by discussing…
The conjecture of Peter Horak and Alex Rosa (generalizing that of Marco Buratti) states that a multiset L of v-1 positive integers not exceeding [v/2] is the list of edge-lengths of a suitable Hamiltonian path of the complete graph with…
In recent developments, a general approach for solving Riemann--Hilbert problems numerically has been developed. We review this numerical framework, and apply it to the calculation of orthogonal polynomials on the real line. Combining this…
We present a common generalization of counting lattice points in rational polytopes and the enumeration of proper graph colorings, nowhere-zero flows on graphs, magic squares and graphs, antimagic squares and graphs, compositions of an…
A fairly general continuation theorem of Leray-Schauder type for the class of so-called admissible multimaps is set forth. This result is then used to establish a universal rule for solving operator inclusions of Hammerstein type in…
We study the connection of the existence of solutions with certain properties and the spectrum of operators in the framework of regular Dirichlet forms on infinite graphs. In particular we prove a version of the Allegretto-Piepenbrink…
In this paper we examine the bi-Hamiltonian structure of the generalized KdV-hierarchies. We verify that both Hamiltonian structures take the form of Kirillov brackets on the Kac-Moody algebra, and that they define a coordinated system.…
In this paper, generalized Pell graphs $\Pi _{n,k}$, $k\ge 2$, are introduced. The special case of $k=2$ are the Pell graphs $\Pi _{n}$ defined earlier by Munarini. Several metric, enumerative, and structural properties of these graphs are…
An important combinatorial result in equivariant cohomology and $K$-theory Schubert calculus is represented by the formulas of Billey and Graham-Willems for the localization of Schubert classes at torus fixed points. These formulas work…
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…
Amongst $d$-regular $r$-uniform hypergraphs on $n$ vertices, which ones have the largest number of independent sets? While the analogous problem for graphs (originally raised by Granville) is now well-understood, it is not even clear what…
A generalization of the well-known Fibonacci sequence is the $k$-Fibonacci sequence with some fixed integer $k\ge 2$. The first $k$ terms of this sequence are $0,0, \ldots, 1$, and each term afterwards is the sum of the preceding $k$ terms.…
We introduce a hypergoemetirc series with two complex variables, which generalizes Appell's, Lauricella's and Kemp\'e de F\'eriet's hypergeometric series, and study the system of differential equations that it satisfies. We determine the…
The concept of partial duality in hypermaps was introduced by Chmutov and Vignes-Tourneret, and Smith independently. This notion serves as a generalization of the concept of partial duality found in maps. In this paper, we first present an…