Combinatorial System: Coefficients, Identities, and Generating Functions

Page | 1 Combinatorial System: Coefficients, Identities, and Generating Functions Chinnaraji Annamalai Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur, India Email: [email protected] https://orcid.org/0000-0002-0992-2584 Abstract: This paper formalizes the unique counting structures introduced by Chinnaraji Annamalai, specifically focusing on what he terms the Annamalai Binomial Coefficient and the resulting power series known as the Combinatorial Geometric Series (CGS). The coefficient is demonstrated to be mathematically equivalent to a widely recognized form of the standard binomial coefficient, often used in problems involving choices with repetition. The study highlights how the CGS, which is constructed through the process of multiple, iterative summations of the basic geometric series, acts as a powerful generating function for this sequence of coefficients. For an infinite series, the resulting closed-form expression is a remarkably simple reciprocal power of the factor (one minus the variable). Furthermore, the framework provides clear formulas for the product of multiple finite geometric series, detailing how a key part of the numerator acts to effectively truncate the infinite series, thereby ensuring the result accurately reflects the finite limits of the original problem. By emphasizing these explicit recursive counting relationships, Annamalai's combinatorial system provides a valuable computational tool for established counting results and offers practical applications in modern technological domains like cybersecurity and machine learning. MSC Classification codes: 05A10, 11B65, 40A05 (65B10) Keywords: computation, binomial identities, binomial series, multiple summations 1. Introduction Combinatorics, the study of discrete structures, relies heavily on fundamental tools such as binomial coefficients [1-5] and generating functions. The standard binomial coefficient ((𝑛 𝑘) is central to counting subsets and coefficients in the binomial expansion [6-9]. Similarly, the geometric series ∑𝑥𝑛=(1−𝑥)−1 is the generating function for the sequence of all ones. This paper examines the binomial structures proposed by Annamalai, which defines a novel binomial coefficient, 𝑉𝑛𝑟, and derives a related power series, the Combinatorial Geometric Series [16-19]. The significance of this framework lies in emphasizing the recursive and product relationships of these coefficients, providing an alternative but equivalent pathway to known results in combinatorial enumeration [20, 21]. 2. Binomial Coefficient Annamalai defines a binomial coefficient 𝑉𝑛𝑟 for non-negative integers 𝑛 and 𝑟. The definition for 𝑉𝑛𝑟 is given by the product form: 𝑉𝑛𝑟=(𝑛+1)(𝑛+2)(𝑛+3)⋯(𝑛+𝑟) 𝑟! =∏𝑛+𝑖 𝑖 𝑟 𝑖=1 Page | 2 The identity between Annamalai's coefficient [10-13] and the standard binomial coefficient is established as follows: 𝑉𝑛𝑟=(𝑛+𝑟)! 𝑛!𝑟! =(𝑛+𝑟 𝑟) Initial conditions are defined as 𝑉0𝑟=𝑉𝑛0=𝑉00=1. A key property of this coefficient is symmetry: 𝑉𝑛𝑟=𝑉𝑟𝑛⇒(𝑛+𝑟 𝑟)=(𝑛+𝑟 𝑛) 3. Combinatorial Geometric Series and Identities The Combinatorial Geometric Series (CGS) is formed by the multiple, iterative summations of the basic geometric series. The 𝑟𝑡ℎ order CGS is defined by the result of 𝑟+1 iterative summations: ∑𝑉𝑖𝑟𝑥𝑖 𝑛 𝑖=0 =∑∑∑⋯ 𝑛 𝑖=0 ⋯⋯∑𝑥𝑖𝑟 𝑛 𝑖=0 𝑛 𝑖=0 𝑛 𝑖=0 ⏟ 𝑟+1 summations Specifically, the 𝑟𝑡ℎ order CGS is defined the following summation structure: ∑𝑉𝑖𝑟𝑥𝑖 𝑛 𝑖=0 =∑ ∑ ⋯⋯⋯∑∑∑𝑥𝑖0 𝑛 𝑖0=0 𝑛 𝑖1=0 𝑛 𝑖2=0 𝑛 𝑖𝑟−1=0 𝑛 𝑖𝑟=0 The 0𝑡ℎ order CGS is the standard geometric series: ∑𝑉𝑖0𝑥𝑖 𝑛 𝑖=0 =∑𝑥𝑖 𝑛 𝑖=0 ,where 𝑉𝑖0=1 Annamalai's binomial theorem states that multiple summations of extended geometric series with binomial coefficients form a binomial series: ∑𝑉𝑖𝑟+1𝑥𝑖= 𝑛 𝑖=0 ∑𝑉𝑖𝑟𝑥𝑖 𝑛 𝑖=0 +∑𝑉𝑖−1 𝑟𝑥𝑖 𝑛 𝑖=1 +∑𝑉𝑖−2 𝑟𝑥𝑖 𝑛 𝑖=2 +⋯⋯⋯+ ∑ 𝑉𝑖−(𝑛−1) 𝑟𝑥𝑖 𝑛 𝑖=𝑛−1 +∑𝑉𝑖−𝑛 𝑟𝑥𝑖 𝑛 𝑖=𝑛 By grouping terms based on the coefficient 𝑉𝑘𝑟 and re-expressing the sums as geometric series, we get: ∑𝑉𝑖𝑟+1𝑥𝑖= 𝑛 𝑖=0 𝑉0𝑟∑𝑥𝑖 𝑛 𝑖=0 +𝑉1𝑟∑𝑥𝑖 𝑛 𝑖=1 +𝑉2𝑟∑𝑥𝑖 𝑛 𝑖=2 +𝑉3𝑟∑𝑥𝑖 𝑛 𝑖=3 +⋯⋯⋯+𝑉𝑛−1 𝑟∑ 𝑥𝑖 𝑛 𝑖=𝑛−1 +𝑉𝑛𝑟∑𝑥𝑖 𝑛 𝑖=𝑛 Substituting the standard formula for the geometric summation,∑𝑥𝑖 𝑛−1 𝑖=𝑘 =𝑥𝑛−𝑥𝑘 𝑥−1 , yields the final series form: Page | 3 ∑𝑉𝑖𝑟+1𝑥𝑖=1 𝑥−1∑𝑉𝑖𝑟𝑥𝑖(𝑥𝑛−𝑖−1) 𝑛−1 𝑖=0 , 𝑛−1 𝑖=0 ∀ 𝑥≠1 A key recursive relationship is that the sum of successive coefficients of order 𝑟 is equal to the next higher-order coefficient, 𝑉𝑛𝑟+1. ∑𝑉𝑖𝑟 𝑛 𝑖=0 = 𝑉0𝑟+𝑉1𝑟+𝑉2𝑟+𝑉3𝑟+⋯+𝑉𝑛−1 𝑟+𝑉𝑛𝑟=𝑉𝑛𝑟+1. 4. Product of Multiple Geometric Series Annamalai's work [14-18] also addresses the product of 𝑟 identical finite geometric series. (∑𝑥𝑖 𝑛−1 𝑖=0 )(∑𝑥𝑖 𝑛−1 𝑖=0 )(∑𝑥𝑖 𝑛−1 𝑖=0 )⋯⋯⋯(∑𝑥𝑖 𝑛−1 𝑖=0 ) ⏟ 𝑟 𝑡𝑖𝑚𝑒𝑠 =(∑𝑥𝑖 𝑛−1 𝑖=0 )𝑟=(1−𝑥𝑛 1−𝑥)𝑟=(1−𝑥𝑛)𝑟 (1−𝑥)𝑟 The binomial expansion of (1−𝑥𝑛)𝑟 is presented as: (1−𝑥𝑛)𝑟= ∑(𝑟 𝑘) 𝑟 𝑘=0 (−𝑥𝑛)𝑘=∑(𝑟 𝑘) 𝑟 𝑘=0 ((−1)𝑥𝑛)𝑘=∑(−1)𝑘(𝑟 𝑘) 𝑟 𝑘=0 𝑥𝑘𝑛 A generalized form of the infinite geometric series is also noted: ∑𝑉𝑖𝑟−1𝑥𝑖= 1 (1−𝑥)𝑟 ∞ 𝑖=0 . 5. Generating Function The generating function for the closed-form expression is given by: 𝑁(𝑥)= ∑(𝑟 𝑘) 𝑟 𝑘=0 (−1)𝑘𝑥𝑘𝑛 =∑(−1)𝑘 𝑟 𝑘=0 𝑉𝑘𝑟−𝑘𝑥𝑘𝑛 =(1−𝑥𝑛)𝑟 The generating function [20, 21] for the infinite sum of the coefficients ∑𝑉𝑖𝑟∞ 𝑖=0 𝑥𝑖 is mentioned below: 𝐺𝑟(𝑥)=∑𝑉𝑖𝑟𝑥𝑖= ∑(𝑖+𝑟 𝑖)𝑥𝑖 ∞ 𝑖=0 ∞ 𝑖=0 This is the generating function for the sequence of coefficients (𝑟 𝑟), (𝑟+1 𝑟), (𝑟+2 𝑟), ⋯⋯ It is a well-known result that the generating function for these coefficients is denoted by: ∑𝑉𝑖𝑟𝑥𝑖= 1 (1−𝑥)𝑟+1 ∞ 𝑖=0 This identity holds as a convergent power series for |𝑥|<1. Page | 4 6. Conclusion Annamalai's work provides a significant extension to classical combinatorial and geometric series theory. By introducing the extended geometric series and the optimized binomial coefficient 𝑉𝑛𝑟, the framework establishes new identities, theorems, and series representations. The derived results, including the Annamalai's Binomial Theorem and the Annamalai Series, along with the understanding of the underlying generative function, are foundational tools that will be useful for future research and development in computational mathematics. References [1] Annamalai, C. (2022) Computation and Calculus for Combinatorial Geometric Series and Binomial Identities and Expansions. The Journal of Engineering and Exact Sciences, 8(7), 14648–01i. https://doi.org/10.18540/jcecvl8iss7pp14648-01i. [2] Annamalai, C. (2022) Application of Factorial and Binomial identities in Information, Cybersecurity and Machine Learning. International Journal of Advanced Networking and Applications, 14(1), 5258-5260. https://doi.org/10.33774/coe-2022-pnx53-v21. [3] Annamalai, C. (2022) Combinatorial and Multinomial Coefficients and its Computing Techniques for Machine Learning and Cybersecurity. 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