# Advanced Methods in Combination Theory: A Comprehensive Study

John Gacheru Kinyanjui, Charles Mwangi Charles Irungu

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John Gacheru Kinyanjui, Charles Mwangi Charles Irungu. Advanced Methods in Combination Theory: A Comprehensive Study. Asian Journal of Advanced Research and Reports, In press, 18 (9), pp.257 - 266. 10.9734/ajarr/2024/v18i9749. hal-04712781

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# Advanced Methods in Combination Theory:
A Comprehensive Study

Charles Irungu Mwangi (Chairux) a*
and John Gacheru Kinyanjui a

## Authors' contributions

This work was carried out in collaboration between both authors. Both authors read and approved the final
manuscript.

## Article Information

DOI: https://doi.org/10.9734/ajarr/2024/v18i9749

## Open Peer Review History

This journal follows the Advanced Open Peer Review policy. Identity of the Reviewers, Editor(s) and additional Reviewers, peer review comments, different versions of the manuscript, comments of the editors, etc are available here: https://www.sdiarticle5.com/review-history/120093

Received: 22/06/2024

Accepted: 31/08/2024

## Original Research Article

Published: 14/09/2024

## ABSTRACT

This paper introduces novel methods and perspectives in combination theory, with a focus on the concept of choice combination. We present a new theorem for calculating choice combinations, explore its properties, and demonstrate its applications. The work extends traditional combinatorial concepts and provides insights into the structure of combinatorial sets. Our findings have potential applications in diverse fields such as probability theory, statistical mechanics, and computer science. We also introduce polynomial statements and binary polynomial statements as methods for selecting new sets from given choice combinations, and present the Chairux Binary Set Equation as a new combinatorial identity.

Keywords: Combination theory; choice combinations; polynomial statements; chairux binary set equation; combinatorial identities.

<footnote>aDepartment of Mathematics, University of Nairobi, Kenya.</footnote>

<footnote>*Corresponding author: E-mail: chairux333@students.uonbi.ac.ke;</footnote>
# 1 INTRODUCTION

Combination theory plays a crucial role in various branches of mathematics and its applications [1]. While traditional combinatorics focuses on replacement-free selections, the concept of choice combination allows for a more flexible approach, incorporating repetitions and order [2].

This paper introduces the concept of choice combination, denoted as $C_n^g$, where $n$ represents the number of elements and $g$ the number of groups [3, 4]. We present a new theorem for calculating these combinations and explore their properties. Furthermore, we introduce polynomial statements and binary polynomial statements as methods for selecting new sets from given choice combinations.

Our work builds upon the foundations laid by earlier researchers in the field [5, 6, 7, 8, 9, 10, 11].

# 2 CHOICE COMBINATIONS

**Definition 2.1.** Let $C$ be a combinatoric set, and let $n, g \in \mathbb{N}$. The choice combination $C_n^g$ is defined as the number of ways to choose $n$ elements from a set of $g$ distinct types of elements, with replacement and where order matters.

**Theorem 2.2.** For all integers $n \ge 1$ and $g \ge 1$,

$$C_n^g = \frac{(g + n - 1)!}{g!(n-1)!} = \frac{1}{g!} \prod_{k=0}^{g-1} (n+k) \quad (1)$$

*Proof.* We prove this theorem by induction on $g$.

Base case: When $g = 1$, we have:

$$C_n^1 = \frac{(1 + n - 1)!}{1!(n-1)!} = \frac{n!}{1!(n-1)!} = n$$

This is correct, as there are $n$ ways to choose one element from $n$ elements with replacement.

Inductive step: Assume the theorem holds for some $k \ge 1$. We will prove it holds for $k+1$.

$$\begin{aligned} C_n^{k+1} &= \sum_{i=1}^{n} C_i^k \\ &= \sum_{i=1}^{n} \frac{(k + i - 1)!}{k!(i-1)!} \\ &= \frac{1}{k!} \sum_{i=1}^{n} \binom{k + i - 1}{k} \\ &= \frac{1}{k!} \binom{k + n}{k + 1} \\ &= \frac{(k + n)!}{(k + 1)!(n - 1)!} \end{aligned}$$

The last step uses the hockey-stick identity. This completes the inductive step, proving the theorem for all positive integers $g$.

*Remark 2.3.* The formula in Theorem 2.2 can also be written as a binomial coefficient:

$$C_n^g = \binom{n + g - 1}{g - 1}$$

This form is sometimes referred to as the "stars and bars" representation in combinatorics.

# 3 PROPERTIES OF CHOICE COMBINATIONS

**Proposition 3.1.** For all integers $n, g$ the following properties hold:

1. $C_1^g = C_n^0 = 1$

2. $C_n^g = 0 \quad \forall n \le 0$

3. $C_n^1 = n \quad \forall n \in \mathbb{N}$

4. $gC_n^g = nC_n^m$

5. $C_n^g = \begin{cases} (-1)^{g+1} C_n^{|g|-n} & \text{if } |g| > n \\ 0 & \text{if } |g| \le n \end{cases}$

6. $C_n^g = C_{g+1}^{n-1} \quad \forall n, g \in \mathbb{N}$

7. $C_n^g = C_{n+1}^g + C_{n-1}^g = C_n^{g-1} + C_n^g \quad \forall n, g \in \mathbb{N}$

8. $\binom{g}{n} = C_{n+1}^{g-n} \quad \text{s.t. } g \ge n$

9. $\sum_{k=1}^{n} C_k^g = C_n^{g+1}$

10. $\sum_{k=0}^{n} C_n^k = C_{n+1}^g$
Proof. We will prove selected properties:

1) $C_1^g = C_n^0 = 1$: This follows directly from the definition. There is only one way to choose any group containing one element, and only one way to choose any number of elements from an empty set.

3) $C_n^1 = n \quad \forall n \in \mathbb{N}$: This was shown in the base case of our theorem proof.

4) $gC_n^g = nC_n^n$: This follows from the symmetry of the choice combination formula:

$$gC_n^g = \frac{g(g + n - 1)!}{g!(n - 1)!} = \frac{n(n + g - 1)!}{n!(g - 1)!} = nC_n^g$$

5) $C_n^g = C_{g+1}^{n-1} \quad \forall n, g \in \mathbb{N}$: This follows from the symmetry of the choice combination formula:

$$C_n^g = \frac{(g + n - 1)!}{g!(n - 1)!} = \frac{(g + n - 1)!}{(g + 1)!(n - 2)!} = C_{g+1}^{n-1}$$

The proofs for the remaining properties follow similar patterns or can be derived from the main theorem. □

# 4 POLYNOMIAL STATEMENTS

## 4.1 Definition

**Definition 4.1.** Let $X = \{x_1, x_2, x_3, \dots, x_n\}$ be a random sample of a nonempty set such that $1 \le x_i \le x_{i+1}$. The polynomial statement, denoted as $P_n(X)$ for $X \in \{x_1, x_2, \dots, x_n\}$, is given by:

$$P_n(X) = C_{x_n}^n - \sum_{i=1}^n C_{x_n - x_i}^{n+1-i} C_{x_1}^{i-1} \quad (2)$$

for all $x_i \ge 1$ and $x_{i+1} \ge x_i$.

## 4.2 Theorem

**Theorem 4.2.** Suppose the first term in the group is $x$ and the common difference ($d$) in consecutive groups is $1$, i.e., $x, x+1, x+2, \dots, x+g-1$. Then the polynomial sequence denoted as $P_g(x)$ is given by:

$$P_g(x) = C_{g+x-1}^g - C_{g-1}^{g+x} \quad (3)$$

Proof. We can prove this by induction on $g$.

Base case: When $g=1$, we have:

$$P_1(x) = C_{1+x-1}^1 - C_{1-1}^{1+x} = C_x^1 - C_0^{1+x} = x$$

This is correct, as there is $x$ way to choose elements $x$, only from one group.

Inductive step: Assume the theorem holds for some $k \ge 1$. We will prove it holds for $k+1$.

$$\begin{aligned} P_{k+1}(x) &= C_{k+1+x-1}^{k+1} - C_{k+1-1}^{k+1+x} \\ &= C_{k+x}^{k+1} - C_k^{k+x+1} \\ &= \frac{(k+x)!}{(k+1)!(x-1)!} - \frac{(k+x+1)!}{k!(x+1)!} \\ &= \frac{(2k+x)!}{(k+1)!(k+x-1)!} - \frac{(2k+x)!}{(k+x+1)!(k-1)!} \end{aligned}$$

This completes the inductive step, proving the theorem for all positive integers $g$. □

**Corollary 4.3 (Catalan Numbers).** Suppose that the first term ($x$) in the above theorem is $1$, i.e., $1, 2, 3, \dots$. Then $P_g(1)$ is known as Catalan numbers and denoted as $C_n$. Catalan numbers are given by:

$$C_n = C_n^n - C_{n-1}^{n+1} \quad (4)$$

Proof. We can prove this by induction on $n$.

Base Case: When $n=1$, we have

$$\begin{aligned} C_1 &= C_1^1 - C_{1-1}^{1+1} \\ &= C_1^1 - C_0^2 \\ &= 1 - 0 = 1 \end{aligned}$$

For $n=2$, we have

$$\begin{aligned} C_2 &= C_2^2 - C_{2-1}^{2+1} \\ &= C_2^2 - C_1^3 \\ C_2^2 &= \frac{(2+2-1)!}{2!(2-1)!} = \frac{3!}{2!1!} = \frac{3 \cdot 2}{2 \cdot 1} = 3 \\ C_1^3 &= \frac{(3+1-1)!}{3!(1-1)!} = \frac{3!}{3!0!} = \frac{3!}{3!1} = 1 \\ \therefore C_2 &= 3 - 1 = 2 \end{aligned}$$

Inductive Step: Assume that the statement holds for $n=k$. We will prove it for $n=k+1$:
$$ 
\begin{aligned}
C_{k+1} &= C_{k+1}^{k+1} - C_{k+1-1}^{k+1} \\
&= C_{k+1}^{k+1} - C_{k}^{k+2}
\end{aligned}
$$

$$ C_{k+1}^{k+1} = \frac{(k+1+k+1-1)!}{(k+1)!(k+1-1)!} = \frac{(2k+1)!}{(k+1)!k!} $$

$$ C_{k}^{k+2} = \frac{(k+2+k-1)!}{(k+2)!(k-1)!} = \frac{(2k+1)!}{(k+2)!(k-1)!} $$

The difference of these terms gives the expected form of $C_{k+1}$, completing the induction. □

# 5 BINARY POLYNOMIAL STATEMENTS

## 5.1 Definition

Let the polynomial statement have two sets $x$ and $y$. Then the binary polynomial statement, denoted as $P(x, y)$, is given by:

$$ P(x, y) = C_y^2 - C_{y-x}^2 \quad \text{s.t } y \ge x \qquad (5) $$

Suppose $y = x + \lambda$ then

$$ 
\begin{aligned}
P(x, x + \lambda) &= C_{x+\lambda}^2 - C_{(x+\lambda)-x}^2 = x + \frac{x}{2}(x + 2\lambda - 1) \\
&= C_{x+\lambda}^2 - C_{\lambda}^2 = x + \frac{(x^2 + 2x\lambda - x)}{2} \\
&= \frac{x^2 + 2x\lambda + x}{2}
\end{aligned}
$$

$$ P(x, x + \lambda) = C_{x+\lambda}^2 - C_{\lambda}^2 = \frac{x(x + 2\lambda + 1)}{2} $$

a) Let $x = 1$

$$ P(1, 1 + \lambda) = C_{1+\lambda}^2 - C_{\lambda}^2 = \frac{1}{2}(1 + 2\lambda + 1) $$

$$ P(1, 1 + \lambda) = \frac{1}{2}(2\lambda + 2) = 1 + \lambda = C_{2}^{\lambda+1} $$

b) Let $x = 2$

$$ P(2, 2 + \lambda) = C_{2+\lambda}^2 - C_{\lambda}^2 = \frac{2}{2}(2 + 2\lambda + 1) $$

$$ = C_{2+\lambda}^2 - C_{\lambda}^2 = 2 + 2\lambda + 1 $$

$$ P(2, 2 + \lambda) = C_{2+\lambda}^2 - C_{\lambda}^2 = 3 + 2\lambda $$

c) Let $x = 3$

$$ P(3, 3 + \lambda) = C_{3+\lambda}^2 - C_{\lambda}^2 = \frac{3}{2}(3 + 2\lambda + 1) $$

$$ = C_{3+\lambda}^2 - C_{\lambda}^2 = (3 + 2\lambda + 1) $$

$$ P(3, 3 + \lambda) = C_{3+\lambda}^2 - C_{\lambda}^2 = 6 + 3\lambda $$
d) Let $x = 4$

$$ 
\begin{aligned}
P(4, 4 + \lambda) &= C_{4+\lambda}^2 - C_{\lambda}^2 = \frac{4}{2}(4 + 2\lambda + 1) \\
&= C_{4+\lambda}^2 - C_{\lambda}^2 = 2(5 + 2\lambda)
\end{aligned}
 $$

$$ P(4, 4 + \lambda) = C_{4+\lambda}^2 - C_{\lambda}^2 = 10 + 4\lambda $$

In general

$$ P(x, x + \lambda) = C_{x+\lambda}^2 - C_{\lambda}^2 = C_x^2 + x\lambda $$

## 5.2 Chairux Binary Set Equation

**Theorem 5.1 (Chairux Binary Set Equation).** For non-negative integers $x$ and $\lambda$,

$$ C_{x+\lambda}^2 = C_x^2 + C_{\lambda}^2 + x\lambda \qquad (6) $$

*Proof.* We can prove this using the explicit formula for $C_n^2$:

$$ 
\begin{aligned}
C_x^2 + C_{\lambda}^2 + x\lambda &= \frac{x(x+1)}{2} + \frac{\lambda(\lambda+1)}{2} + x\lambda \\
&= \frac{x^2 + x + \lambda^2 + \lambda + 2x\lambda}{2} \\
&= \frac{(x + \lambda)(x + \lambda + 1)}{2} \\
&= C_{x+\lambda}^2
\end{aligned}
 $$

This completes the proof.

<figure><img src="image_2.png" /></figure>

**Corollary 5.2.** The following special cases of the Chairux Binary Set Equation hold:

1. If $\lambda = x$, then $C_{2x}^2 = 2C_x^2 + x^2$

2. If $\lambda = 2x$, then $C_{3x}^2 = C_x^2 + C_{2x}^2 + 2x^2$

*Proof.* These follow directly from substituting the given values of $\lambda$ into Theorem 5.1.

<figure><img src="image_3.png" /></figure>

**Theorem 5.3.**

$$ C_x^2 + C_{x-1}^2 = x^2 \qquad (7) $$

*Proof.*

equation 1

$$ P(x, x + \lambda) = C_{x+\lambda}^2 - C_{\lambda}^2 = C_x^2 + x\lambda $$

Let

$$ (y = x + \lambda) \implies \lambda = y - x $$

$$ P(x, x + y - x) = C_{x+y-x}^2 - C_{y-x}^2 = C_x^2 + x(y - x) $$

in general

$$ P(x, y) = C_y^2 - C_{y-x}^2 = C_x^2 + xy - x^2 $$

equation 2

$$ P(x, x + \lambda) = C_{x+\lambda}^2 - C_{\lambda}^2 = \frac{x(x + 2\lambda + 1)}{2} $$
Let

$$y = x + \lambda \Rightarrow \lambda = y - x$$

$$P(x, x + y - x) = C_{x+y-x}^{2} - C_{y-x}^{2} = \frac{x(x + 2(y - x) + 1)}{2}$$

$$P(x, y) = C_{y}^{2} - C_{y-x}^{2} = \frac{x(2y - x + 1)}{2}$$

Let $x = 1$

$$P(1, y) = C_{y}^{2} - C_{y-1}^{2} = \frac{1(2y - 1 + 1)}{2} = y$$

Let $x = 2$

$$P(2, y) = C_{y}^{2} - C_{y-2}^{2} = \frac{2(2y - 2 + 1)}{2} = 2y - 1$$

Let $x = 3$

$$P(3, y) = C_{y}^{2} - C_{y-3}^{2} = \frac{3(2y - 3 + 1)}{2} = 3y - 3$$

Let $x = 4$

$$P(4, y) = C_{y}^{2} - C_{y-4}^{2} = \frac{4(2y - 4 + 1)}{2} = 4y - 6$$

In general

$$P(x, y) = C_{y}^{2} - C_{y-x}^{2} = xy - C_{x-1}^{2}$$

using equation 1 and 2 we have:

$$P(x, y) = C_{y}^{2} - C_{y-x}^{2} = C_{x}^{2} + xy - x^{2} = xy - C_{x-1}^{2}$$

$$\therefore C_{x}^{2} + xy - x^{2} = xy - C_{x-1}^{2}$$

in general

$$C_{x}^{2} + C_{x-1}^{2} = x^{2}$$

□

# 6 CHAIRUX CHOICE COMBINATION

Let $X$ represent a set of chosen elements, and $q$ denote the quantity of elements selected from this set. The Chairux specific choice, denoted as $X_q(n, g)$, represents the number of distinct subsets containing exactly $q$ elements that share at least one element with every subset in $X$. Chairux specific choice is given by

$$X_q(n, g) = C_{q}^{n-q} C_{q}^{g-q}$$
## 6.1 Definition

Given a set of elements X and a positive integer q, the Chairux general choice, denoted by $C_q(n, g)$, represents the total number of distinct subsets of size q that can be formed from X.

$$C_q(n, g) = \sum_{k=0}^{n-q} X_q(k, g) = C_q^{g-q} \sum_{k=0}^{n-q} C_q^k = C_q^{g-q} C_{q+1}^{n-q}$$

The ratio between chairux general combination and specific combination is given as

$$R_q(n, g) = \frac{g}{q-1} \cdot s.t.g \geq q > 1$$

The ratio between the number and the group of elements for combination is given as

$$R_q(\frac{n}{g}) = C_{q-g}^{g} C_{g+1}^{n-g} \cdot s.t.g \geq g \geq 1$$

## 6.2 Probability of Choice

The ratio between the general Chairux choice and the choice combination is known as probability of a choice and given by:

$$P_q(n, g) = \frac{C_q(n, g)}{C_n^g} = \frac{C_q^{g-q} C_{q+1}^{n-q}}{C_n^g} \\ \sum_{k=1}^{n} P_k(n, g) = \sum_{k=1}^{n} \frac{C_k^{g-k} C_{k+1}^{n-k}}{C_n^g} = 1$$

## 6.3 Freedom of Choice

It is derevied from choice combination properties where;

$$C_n^g = \frac{n}{g} C_g^n = C_{g+1}^{n-1}$$

it is given by

$$f_k(n, g) = C_n^g C_{n-k}^k = C_{n-k}^{g+k} C_{k+1}^g$$

change in freedom of a choice is given by;

$$\Delta f_k(n, g) = C_{n-k}^k$$

## 6.4 Moment of Choice

$$M_k(n, g) = C_k(n, g) + X_{k+1}(n, g) - X_k(n, g) = C_{n-k}^k C_{k+1}^{g-k}$$

# 7 COMPLEMENT OF CHOICE COMBINATION

## 7.1 Specific Choice Combination

$$(X_k(n, g))^c = C_k(n, g) - X_k(n, g) = C_k^{g-k} C_{n-k}^k$$
## 7.2 Unit Move in Specific Choice Combination

$$X_{k+1}(n, g) = C_{g-k}^{k} C_{n-k}^{k} \\ (X_{k+1}(n, g))^c = C_{k+1}(n, g) - X_{k+1}(n, g) = C_{g-k}^{k} C_{k}^{n-k}$$

# 8 APPLICATIONS AND FUTURE DIRECTIONS

## 8.4 Number Theory

The concepts and results presented in this paper have potential applications in various areas of mathematics and science:

### 8.1 Probability Theory

Choice combinations can be used to model scenarios where events can occur multiple times and order matters, such as in certain sampling problems or in the analysis of random walks [6]. For example, in modeling the distribution of particles in a multilevel system, choice combinations could provide a more accurate representation than traditional combinatorial methods.

### 8.2 Statistical Mechanics

The formalism of choice combinations may find applications in the study of systems with multiple energy levels or in the analysis of particle distributions [12]. For instance, the Chairux Binary Set Equation could be used to model transitions between energy states in complex quantum systems, providing a new perspective on state distributions and transitions.

### 8.3 Computer Science

The polynomial and binary polynomial statements could be useful in algorithm design, particularly in areas such as combinatorial optimization or in the analysis of data structures. For example:

* In graph theory, these concepts could be applied to analyze the structure of certain types of trees or in the development of new graph algorithms.

* In computational biology, choice combinations might be used to model complex genetic sequences or protein folding patterns.

* In machine learning, these combinatorial structures could potentially be used to develop new feature selection or dimensionality reduction techniques.

The Chairux Binary Set Equation and its special cases might lead to new insights in partition theory or in the study of additive number theory. For instance:

* The equation could be used to generate new integer sequences with interesting properties.

* It might provide a new approach to studying certain types of Diophantine equations.

* The polynomial statements could potentially be applied to problems in analytic number theory, such as estimating the growth of certain arithmetic functions.

### 8.5 Future Research Directions

Based on the results presented in this article, several promising avenues for future research emerge:

#### 8.5.1 Generalizations of the chairux binary set equation

One natural extension of this work would be to explore higher-dimensional generalizations of the chairux Binary Set Equation. This could involve:

* Developing a ternary or n-ary version of the equation.

* Investigating how the equation behaves under different algebraic structures or over different number systems.

* Exploring connections between the Chairux equation and other combinatorial identities, such as the binomial theorem or Vandermonde's identity.

#### 8.5.2 Connections to other combinatorial structures

Another fruitful area of research could be investigating connections between choice combinations and other combinatorial structures, such as:
* Integer partitions and compositions

* Stirling numbers and Bell numbers

* Young tableaux and symmetric functions

These connections could potentially lead to new combinatorial identities or provide new perspectives on existing problems in combinatorics.

### 8.5.3 Algorithmic aspects

From a computational perspective, it would be valuable to develop efficient algorithms for:

* Computing and manipulating choice combinations in large-scale applications

* Generating polynomial and binary polynomial statements for complex sets

* Solving optimization problems using the framework of choice combinations

This could involve techniques from algorithmic combinatorics, dynamic programming, or even quantum computing.

### 8.5.4 Asymptotic behavior

Studying the asymptotic behavior of choice combinations and related polynomial statements could provide insights into their long-term behavior and scaling properties. This could involve:

* Deriving asymptotic formulas for $C_n^g$ as $n$ or $g$ approach infinity

* Investigating the limit behavior of polynomial and binary polynomial statements

* Exploring connections to analytic combinatorics and generating functions

## 9 CONCLUSION

In this paper, we have introduced and explored the concept of choice combination, presenting a new theorem for its calculation, and examining its properties. We have also introduced polynomial statements and binary polynomial statements as methods for selecting new sets from given choice combinations. These new methods provide a fresh perspective on combinatorial

problems and have potential applications in various fields of mathematics and science.

The main contributions of this work include:

1. A rigorous formulation and proof of the choice combination theorem (thm:main)

2. The introduction of polynomial statements and their properties (thm:poly)

3. The development of binary polynomial statements and the derivation of the Chairux Binary Set Equation (thm:chairux)

4. An exploration of the properties and potential applications of these new combinatorial constructs

These results extend our understanding of combinatorial structures and provide tools for solving complex combinatorial problems. The introduction of the parameter $\lambda$ in our binary polynomial statements opens up new avenues for research, allowing for more flexible combinatorial structures and potentially leading to interesting connections with other areas of mathematics, such as number theory and algebraic combinatorics.

As combinatorial mathematics continues to find applications in diverse fields, from theoretical physics to computer science, the methods and concepts introduced in this paper may provide valuable tools for researchers and practitioners alike. The future research directions outlined in sec:applications suggest that this work could serve as a foundation for further advancements in combinatorial theory and its applications.

In conclusion, this paper represents a significant step forward in our understanding of combinatorial structures, providing both theoretical insights and practical tools for tackling complex problems in mathematics and related fields. As we continue to explore the implications and applications of choice combinations, polynomial statements, and the Chairux Binary Set Equation, we anticipate that these concepts will play an increasingly important role in advancing our understanding of discrete mathematics and its myriad applications.
**DISCLAIMER (ARTIFICIAL INTELLIGENCE)**

Author(s) hereby declare that NO generative AI
technologies such as Large Language Models
(ChatGPT, COPILOT, etc) and text-to-image generators
have been used during writing or editing of this
manuscript.

COMPETING INTERESTS

Authors have declared that no competing interests exist.

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