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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.

There are one-to-one relationships between binomial coefficients and PASCAL triangles. Mathematicians from ancient to modern are deeply interested with them from the angle of in variances [

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 [

Binomial coefficient [

Combinatorial number [

The variant construction [

Based on the traditional vector state, the system extends its phase spaces using two kinds of vector operations: permutation and complementarity. The n logical function space is extended from a traditional function space to a functional configuration space by two extended logic operations.

Variant measurements and variant maps have been applied to different problems, including classical cryptographic sequence analysis, quantum cryptographic sequence detection, chromosome whole gene sequence illustration, electrocardiogram signal detection, bat signal processing and other applications [

This paper focuses attention mainly to extend a basic binomial formula in variant measurement [

The reflection transformation [

Rotation invariant [

Ordered structure of four measures: {a, b, c, d} in the monograph on variant constructions [

In this paper, the binomial formula f ( m , p , k ) = ( m − p k ) ( p k ) is selected, and the value range is 0 ≤ p ≤ m; 0 ≤ k ≤ p; k ≤ m − p.

According to the binomial coefficient formula, multiple steps of transformations on different values contain various features of reflection and rotation invariances. Different binomial coefficient formulas are shown in four regions with distinguished characteristics. On a set of four binomial coefficient formula, their parameters and valued ranges are different. The transforming process for the four Zones {I, II, III, IV} is shown in

The Zone I can be calculated from the formula. Range of Zone III can be carried out by reflecting binomial coefficients of corresponding positions in

Because of reflection and rotation invariance between Zone I, III and Zone II, IV, rotated from Zone I, III to Zone II, IV. The binomial coefficients of the four regions are symmetric with the diagrams on m − p = k and p = k.

In

1) f ( m , p , k ) = ( m − p k ) ( p k ) , 0 < p < m; 0 < k < p; k < m − p;

2) f 3 ( m , p , k ) = ( m − k p ) ( k p ) , 0 ≤ k ≤ m; 0 ≤ p ≤ k; k ≤ m − p;

3) f 2 ( m , p , k ) = ( k m − p ) ( m − k m − p ) , 0 ≤ p ≤ m; 0 ≤ k ≤ p; k > m − p;

4) f 4 ( m , p , k ) = ( p m − k ) ( m − p m − k ) , 0 ≤ k ≤ m; 0 ≤ p ≤ k; k > m − p.

In three-dimensional representation:

The distribution of different values is analyzed and discussed visually by using the measurement diagram in the combinatorial coefficients.

Applying the function f_{1} m = 11, a total of 42 points are in the region of 12*12 shown in

When m = 11 and m = 10, the three-dimensional visualization results of function f1 are shown in Figures 3(a)-(d).

For Zone II, m = 11 and m = 10 as examples to compare the coefficient distribution by the rule of odd-even after reflection binomial formula.

Finally, for Zone II and IV, rotation effects are illustrated by the extended binomial coefficient distributions shown in Figures 4(a)-(d) through f_{2} and f_{4}, as the rotation of f_{1} and f_{3} functions on rotation invariant.

There is no vertex at the top of the triangle on the top view, and the highest value is represented for two numerical points when m is odd. The top view triangle has the highest vertex when m is even.

It is a novel approach to explore rotation and reflection patterns using the four binomial coefficient formulas to show their coefficient distribution characteristics as reflection and rotation transformations. It is important to distinguish reflection and rotation characteristics from the selected coefficient binomial formulas and sample distributions following the condition of parameter transformations on

odd and even numbers. A series of sample results are illustrated to explore the potential expansion ability of variant constructions to form a set of new binomial formula. It lays a theoretical foundation for the basic and applied research on big data analysis and simulation targets with complex combinatorial measurements in the future.

This project is supported by The Key Project of Quantum Communication Technology of Yunnan (2018ZI002) and Yunnan overseas high-level Scholar project.

The authors declare no conflicts of interest regarding the publication of this paper.

Zhu, M.H. and Zheng, J. (2019) Research on Transformation Characteristics of Binomial Coefficient Formula and Its Extended Model. Journal of Applied Mathematics and Physics, 7, 2927-2932. https://doi.org/10.4236/jamp.2019.711202