Related papers: Generating Functions for Multi-j-Symbols
We have the generating function which determines the number of $2$-bridge knot groups admitting epimorphisms onto the knot group of a given $2$-bridge knot, in terms of crossing number. In this paper, we will refine this formula by taking…
In this paper the number of ways to glue together several polygons into a surface of genus $g$ has been investigated. We've given an elementary proof on the formula for the generating function $\mathbf{C}_g^{[2]}(z)$ of the number of…
Analogues of classical combinatorial identities for elementary and homogeneous symmetric functions with coefficients in Yanigian are discussed. As a corollary, similar relations are deduced for shifted Schur functions.
We conjecture a closed form expression for the simplest class of multiplicity-free quantum 6j-symbols for U_q(sl_N). The expression is a natural generalization of the quantum 6j-symbols for U_q(sl_2) obtained by Kirillov and Reshetikhin.…
We give a combinatorial interpretation for the hypergeometric functions associated with tuples of rational numbers.
When all the elements of the multiset $\{1,1,2,2,3,3,\ldots,k,k\}$ are placed in the cells of a $2\times k$ rectangular array, in how many configurations are exactly $v$ of the pairs directly over top one another, and exactly $h$ directly…
Multimodal normal incestual systems are investigated in terms of multiple categories. The different sorted composition of operators are exhibited as 2-cells in multiple categories built up from 2-categories giving rise to different axioms.…
We compute some 6j symbols in the category of polynomial representations of $GL(\infty)$.
By introducing a class of meromorphic functions with certain ramification structures on $\Bbb{CP}^1$, a new method for the determination of the Legendre representation of elliptic curves with complex multiplication is introduced. These…
We characterise the quartic (i.e. 4-regular) multigraphs with the property that every edge lies in a triangle. The main result is that such graphs are either squares of cycles, line multigraphs of cubic multigraphs, or are obtained from…
We classify and construct all line graphs that are $3$-polytopes (planar and $3$-connected). Apart from a few special cases, they are all obtained starting from the medial graphs of cubic (i.e., $3$-regular) $3$-polytopes, by applying two…
It is well-known that differentiation of hypergeometric function multiplied by a certain power function yields another hypergeometric function with a different set of parameters. Such differentiation identities for hypergeometric functions…
Using a general $q$-series expansion, we derive some nontrivial $q$-formulas involving many infinite products. A multitude of Hecke--type series identities are derived. Some general formulas for sums of any number of squares are given. A…
There are a number of ways to procedurally generate interesting three-dimensional shapes, and a method where a cellular neural network is combined with a mesh growth algorithm is presented here. The aim is to create a shape from a genetic…
The article contains some important classes of multisets. Combinatorial proofs of problems on the number of m-submultisets and m-permutations of multiset elements are considered and effective algorithms for their calculation are given. In…
Extended double shuffle relations for multiple zeta values are obtained by the fact that any product of regularized multiple zeta values has two different representations, and the case of two-fold product is considered in general. In this…
In this paper geometry is studied with a novel approach. Every geometrical object is defined as a symbol which satisfies some properties. These symbols are then coded into a class of numbers which are named here as many dots numbers (MDN).…
In this paper, cylindric partitions into profiles $c=(1,1)$ and $c=(2,0)$ are considered. The generating functions into unrestricted cylindric partitions and cylindric partitions into distinct parts with these profiles are constructed. The…
Music is inherently made up of complex structures, and representing them as graphs helps to capture multiple levels of relationships. While music generation has been explored using various deep generation techniques, research on…
One of the main purposes of this article is to give functional equations and differential equations between Bernstein basis functions and generating functions of B-spline curves. Using these equations, very useful formulas containing the…