Abstract
Combinatorial methods are used to give specific mathematics, and the proof of combinatorial identities is the hot spot research of combinatorial mathematics. Binomial coefficients play an important role in the fields of physics, mathematics and computer science. In variances of rotation and reflection is the core characteristic for combinatorial systems. This paper uses different parameters to form reflection, rotation and specific features by using variant construction model and method, meanwhile the quantitative distribution characteristics of binomial coefficient formula are analyzed by three-dimensional maps. Using variant construction, the combinatorial clustering properties are investigated to apply binomial formulas and sample distributions, and various combinatorial patterns are illustrated. It is proved that the basic binomial coefficient formula and its extended model have obvious properties of reflection and rotation invariance.
Highlights
There are one-to-one relationships between binomial coefficients and PASCAL triangles
This paper uses different parameters to form reflection, rotation and specific features by using variant construction model and method, the quantitative distribution characteristics of binomial coefficient formula are analyzed by three-dimensional maps
It is proved that the basic binomial coefficient formula and its extended model have obvious properties of reflection and rotation invariance
Summary
There are one-to-one relationships between binomial coefficients and PASCAL triangles. The resulting distribution with the family of that binomial formula and its extended model contain obvious invariant properties of reflection and rotation in the paper. It is expected that this kind of invariant formula can be closely associated with the invariant properties in quantum mechanics, classical physics, mathematics, computer science and convolutional neural network, which could be helpful to explore the optimization research of various applied computing methods [3]
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