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Math Functions in Calculator

The Calculator class in JNumberTools provides efficient, thread-safe mathematical functions for combinatorics and number theory computations. It uses memoization for performance, supporting operations like factorials, binomial coefficients, multinomial coefficients, and more, with BigInteger compatibility for large-scale calculations.

Key Features:

• Thread-safe with synchronized caches for nCr, nPr, factorial, and subfactorial values.
• Optimized for both small and large inputs using memoization and BigInteger.
• Includes utility methods for combinatorial counts, number bounds, and basic arithmetic operations.

Currently Available Functions

1. Factorial and Subfactorial
   1. Factorial
   2. Subfactorial
   3. Factorial Upper Bound
2. Combinations and Permutations
   1. nCr (Binomial Coefficient)
   2. nCr with Repetition
   3. nCr Upper Bound
   4. nPr (Permutation Coefficient)
3. Multiset Calculations
   1. Multinomial Coefficient
   2. Total m-th Multinomial
   3. Multiset Combinations Count All
   4. Multiset Combinations Count
   5. Multiset Combination Count starting from index
4. Subset Calculations
   1. Total Subsets in Range
5. Arithmetic Operations
   1. Power
   2. Greatest Common Divisor (GCD)
   3. Least Common Multiple (LCM)
6. Special Combinatorial Functions
   1. Rencontres Number

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1. Factorial and Subfactorial

1.1 Factorial

Computes the factorial of a non-negative integer n (n!), using a memoized cache for efficiency.

Calculator calc = new Calculator();
// Computes 5! = 120
BigInteger result = calc.factorial(5);
System.out.println(result); // Output: 120

1.2 Subfactorial

Calculates the subfactorial (!n), the number of derangements of n items, using memoization.

Calculator calc = new Calculator();
BigInteger result = calc.subFactorial(4);
System.out.println(result); // Output: 9

1.3 Factorial Upper Bound

Finds the smallest n such that n! exceeds a given value.

Calculator calc = new Calculator();
int result = calc.factorialUpperBound(BigInteger.valueOf(1000));
System.out.println(result); // Output: 7

2. Combinations and Permutations

2.1 nCr (Binomial Coefficient)

Calculates nCr, the number of ways to choose r items from n without repetition, using memoization.

Calculator calc = new Calculator();
BigInteger result = calc.nCr(5, 2);
System.out.println(result); // Output: 10

2.2 nCr with Repetition

Computes the binomial coefficient with repetition allowed, equivalent to C(n + r - 1, r).

Calculator calc = new Calculator();
BigInteger result = calc.nCrRepetitive(4, 3);
System.out.println(result); // Output: 20

2.3 nCr Upper Bound

Finds the smallest n such that nCr(n, r) exceeds a given value.

Calculator calc = new Calculator();
int result = calc.nCrUpperBound(3, BigInteger.valueOf(100));
System.out.println(result); // Output: 9

2.4 nPr (Permutation Coefficient)

Calculates nPr, the number of ways to arrange r items from n, using memoization.

Calculator calc = new Calculator();
BigInteger result = calc.nPr(5, 3);
System.out.println(result); // Output: 60

3. Multiset Calculations

3.1 Multinomial Coefficient

Computes the multinomial coefficient for given counts, representing distinct permutations of a multiset.

Calculator calc = new Calculator();
BigInteger result = calc.multinomial(2, 1, 3);
System.out.println(result); // Output: 60

3.2 Total m-th Multinomial

Calculates the total number of m-th permutations for a multiset, given a start index and step m.

Calculator calc = new Calculator();
BigInteger result = calc.totalMthMultinomial(0, 2, 2, 1, 3);
System.out.println(result); // Output: 30

3.3 Multiset Combinations Count All

Computes the number of ways to select exactly s items from a multiset for all s in the range [0, ⌊total⌋/2], where total is the sum of the frequencies. Returns an array of counts for selecting 0 to ⌊total/2⌋ items, using dynamic programming.

int[] result = Calculator.multisetCombinationsCountAll(2, 1);
System.out.println(Arrays.toString(result)); // Output: [1, 2, 1]

3.4 Multiset Combinations Count

Calculates the number of ways to select exactly k items from a multiset.

BigInteger result = Calculator.multisetCombinationsCount(2, 2, 1);
System.out.println(result); // Output: 2

3.5 Multiset Combination Count starting from given index

Counts the number of ways to select exactly k items from a multiset defined by frequencies, considering only the item types from the specified index onward.

int k = 5;
int index = 2;
int[] frequencies = {10, 12, 5, 8};
BigInteger result = Calculator.multisetCombinationsCountStartingFromIndex(k, index, frequencies);

4. Subset Calculations

4.1 Total Subsets in Range

Computes the total number of subsets of sizes within a given range from n elements.

Calculator calc = new Calculator();
BigInteger result = calc.totalSubsetsInRange(1, 2, 4);
System.out.println(result); // Output: 10

5. Arithmetic Operations

5.1 Power

Calculates base raised to exponent, supporting both long and BigInteger inputs.

Calculator calc = new Calculator();
BigInteger result = calc.power(2, 10);
System.out.println(result); // Output: 1024

5.2 Greatest Common Divisor (GCD)

Computes the GCD of multiple BigInteger numbers using the binary GCD algorithm.

Calculator calc = new Calculator();
BigInteger result = calc.gcd(BigInteger.valueOf(48), BigInteger.valueOf(18), BigInteger.valueOf(27));
System.out.println(result); // Output: 3

5.3 Least Common Multiple (LCM)

Computes the LCM of multiple BigInteger numbers using a hybrid iterative/recursive approach.

Calculator calc = new Calculator();
BigInteger result = calc.lcm(BigInteger.valueOf(4), BigInteger.valueOf(6), BigInteger.valueOf(8));
System.out.println(result); // Output: 24

6. Special Combinatorial Functions

6.1 Rencontres Number

Calculates the number of permutations of n items with exactly k fixed points.

Calculator calc = new Calculator();
BigInteger result = calc.rencontresNumber(4, 1);
System.out.println(result); // Output: 8

Home </br>Permutation Generators </br>Combination Generators </br>Set/Subset Generators </br>Cartesian Product Generators </br>Math Functions </br>Ranking Algorithms </br>Number System Algorithms

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