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A Method for Obtaining Generating Function for Central Coefficients of Triangles
Obtaining Generating Function Central Coefficients of Triangles Combinatorics
2012/6/21
We consider problems of obtaining a generating function for the central coefficients of triangle $T(n,k)$, which is given by expression $[xG(x)]^k=\sum_{n\geqslant k} T(n,k)x^n$, $G(0)\neq 0$. We prov...
This note deals with the relationship between the total number of $k$-walks in a graph, and the sum of the $k$-th powers of its vertex degrees. In particular, it is shown that the the number of all $k...
The set of Dyck paths of length $2n$ inherits a lattice structure from a bijection with the set of noncrossing partitions with the usual partial order. In this paper, we study the joint distribution o...
Determination of Integral Cayley Graphs on Finite Abelian Groups
abelian group character cayley graph integral graph
2012/6/21
A graph is integral means that all its eigenvalues are integers. In this note, we determine all the integral Cayley graphs on finite abelian groups. Moreover, we calculate the the number of integral C...
We prove that any distributional limit of finite planar graphs in which the degree of the root has an exponential tail is almost surely recurrent. As a corollary, we obtain that the uniform infinite p...
We present a new approach for counting trees, and we apply it to count multitype Cayley trees and to prove the multivariate Lagrange inversion formula. The gist of our approach is to exploit the symme...
Enumerations of finite topologies associated with a finite graph
Enumerations of finite topologies finite graph Combinatorics
2012/6/19
The number of topologies and non-homeomorphic topologies on a fixed finite set are now known up to $n=18$, $n=16$ but still no complete formula yet (Sloane). There are one to one correspondence among ...
Mutually unbiased bases as submodules and subspaces
Mutually unbiased bases submodules and subspaces Combinatorics
2012/6/19
Mutually unbiased bases (MUBs) have been used in several cryptographic and communications applications. There has been much speculation regarding connections between MUBs and finite geometries. Most o...
Polytope numbers for a polytope are a sequence of nonnegative integers that are defined by the facial information of a polytope. Every polygon is triangulable and a higher dimensional analogue of this...
The Sorting Index and Permutation Codes
Permutation statistics Mahonian statistics Coxeter groups Set-valued statistics Bijections
2012/6/19
In the combinatorial study of the coefficients of a bivariate polynomial that generalizes both the length and the reflection length generating functions for finite Coxeter groups, Petersen introduced ...
Let $v(n)$ be the minimum number of voters with transitive preferences which are needed to generate any strong preference pattern (ties not allowed) on $n$ candidates. Let $k=\lfloor \log_2 n\rfloor$....
The characteristic imset polytope for diagnosis models
graphical model characteristic imset polytope diagnosis model Bipartite graph
2012/6/15
In 2010, M. Studen\'y, R. Hemmecke, and Linder explored a new algebraic description of graphical models, characteristic imsets. Compare with standard imsets, characteristic imsets have several advanta...
Eigenvalue interlacing and weight parameters of graphs
Eigenvalue interlacing weight parameters of graphs Combinatorics
2012/6/15
Eigenvalue interlacing is a versatile technique for deriving results in algebraic combinatorics. In particular, it has been successfully used for proving a number of results about the relation between...
Let A be the adjacency matrix of a graph $X$ and suppose U(t)=exp(itA). We view A as acting on $\cx^{V(X)}$ and take the standard basis of this space to be the vectors $e_u$ for $u$ in $V(X)$. Physici...
Expected Patterns in Permutation Classes
Expected Patterns Permutation Classes Combinatorics
2012/6/15
In the set of all patterns in $S_n$, it is clear that each k-pattern occurs equally often. If we instead restrict to the class of permutations avoiding a specific pattern, the situation quickly become...