Concepts158
Bayes' Theorem
Bayes' Theorem tells you how to update the probability of a hypothesis after seeing new evidence.
Partition Function
The partition function p(n) counts the number of ways to write n as a sum of positive integers where order does not matter.
Generating Functions - OGF
An ordinary generating function (OGF) encodes a sequence (a_n) as a formal power series A(x) = \sum_{n \ge 0} a_n x^n.
Stirling Numbers of First Kind
Stirling numbers of the first kind count permutations by their number of cycles and connect power polynomials to rising/falling factorials.
Lucas' Theorem
Lucas' Theorem lets you compute C(n, k) modulo a prime p by working digit-by-digit in base p.
Inclusion-Exclusion Principle
The Inclusion-Exclusion Principle (IEP) corrects overcounting by alternately adding and subtracting sizes of intersections of sets.
Linear Diophantine Equations
A linear Diophantine equation ax + by = c has integer solutions if and only if gcd(a, b) divides c.
Binomial Theorem and Identities
The binomial theorem expands (x + y)^n into a sum of terms using binomial coefficients that count how many ways to choose k items from n.
Discrete Logarithm
The discrete logarithm problem asks for x such that g^x ≡ h (mod p) in a multiplicative group modulo a prime p.
Pollard's Rho Factorization
Pollard's Rho is a randomized algorithm that finds a non-trivial factor of a composite integer by walking a pseudorandom sequence modulo n and extracting a factor with a gcd.
Miller-Rabin Primality Test
Miller–Rabin is a fast primality test that uses modular exponentiation to detect compositeness with very high reliability.
Permutations and Combinations
Permutations count ordered selections, while combinations count unordered selections.