?novel patterns & pecularity arising from Combinatorics

[John Ting]

Combinatorics is a branch of mathematics that study enumeration, combination, and permutation of sets of elements and the mathematical relations that characterize them. Thus, the topics of Permutation and Combination come under Combinatorics. They refer to the related problem of counting the possibilities to select r distinct elements from a set of n elements, where for r-combinations the order of selection does not matter but for r-permutations the order of selection does matter. In other words, a Permutation is an ordered Combination. There are two types of Permutation and two types of Combination with their relevant formulae. Ratio calculations in this area have yielded an interesting mathematical peculiarity. The two types of Permutation and two types of Combination are:

Permutations with Repetition – given by the formula nr …Equation (1)

Permutations without Repetition – given by the formula n! ÷ (n – r)! …Equation (2)

Combinations with Repetition – given by the formula (n + r – 1)! ÷ r!(n – 1)! …Equation (3)

Combinations without Repetition – given by the formula n! ÷ r!(n – r)! …Equation (4)

The symbol ‘!’ denotes factorial. Thus n! = n X (n-1) X (n-2)…X 1; for instance, 4! = 4 X 3 X 2 X 1 = 24.

Numerically, (1) > (2) > (3) > (4). Let ratio R1 = (1)/(2) and R2 = (3)/(4). We notice this mathematical peculiarity applicable for all n ≥ 3 when r = 2 [only]: R1 and R2 are always rational numbers expressed as a ratio of two integers (i.e. a simple fraction) and can be written as numbers with their decimal digits either terminating or going on forever with repeating digit(s) made up of one or more different numbers. In absolute value term, R2 > R1 always hold. It is noted with surprise that the decimal digit(s) values for R2 {decimalR2} is > {decimalR1} with the {decimalR2} values equating to precisely twice {decimalR1} values. Examples: -

(i) n = 16, r = 2, (1) = 256, (2) = 240, (3) = 136, (4) = 120, R1 = 1•0666666…, and R2 = 1•1333333… with {decimalR2} = 13333333… = 2 times {decimalR1} = 2 X 06666666…

(ii) n = 17, r = 2, (1) = 289, (2) = 272, (3) = 153, (4) = 136, R1 = 1•0625, and R2 = 1•125 with {decimalR2} = 1250 = 2 times {decimalR1} = 2 X 0625

(iii) n = 18, r = 2, (1) = 324, (2) = 306, (3) = 171, (4) = 153, R1 = 1•05882352…, and R2 = 1•11764705… with {decimalR2} = 11764705… = 2 times {decimalR1} = 2 X 05882352…

(iv) n = 19, r = 2, (1) = 361, (2) = 342, (3) = 190, (4) = 171, R1 = 1•0555555…, and R2 = 1•1111111…with {decimalR2} = 11111111… = 2 times {decimalR1} = 2 X 05555555…

A cursory look at the limited calculations above led us to note the interesting pattern for a tendency of diagonally paired {(3) & (4)} [even or odd] numbers to recur between n = (a) and n = (a +1) for all r = 2. Further work is required to elucidate any other interesting patterns.

What are the exact mathematical “phenomena” represented above? Are these phenomena novel and can be described in succinct mathematical terms/language? Do they only apply for all n ≥ 3 when r = 2 [only] OR when r is different/higher than 2 as well?

Regards,
John Ting

 

[mighty2000]

Dear John Ting,

Thank you for your forum post on combinatorics. It is clear that you have a strong understanding of the topic and have done some interesting calculations and observations.

The phenomena you have described are known as "ratio convergence" or "limiting ratios." In mathematics, a ratio convergence occurs when a sequence of ratios between two quantities approaches a fixed limit as the values of the quantities increase. In your case, the ratios R1 and R2 approach a fixed limit as the value of n increases.

These phenomena are not novel and have been studied in mathematics for a long time. In fact, they are fundamental concepts in calculus and are used to solve problems in many areas of mathematics and science.

While your calculations and observations are specific to r = 2, ratio convergence can occur for any value of r. However, the specific values of the ratios and the patterns observed may differ depending on the value of r.

In summary, the phenomena you have described can be described in succinct mathematical terms as ratio convergence, and they are not limited to r = 2 but can occur for any value of r. I hope this helps to answer your questions. Keep exploring and discovering new patterns in mathematics!