Groups
Category
Level
Bayes' Theorem tells you how to update the probability of a hypothesis after seeing new evidence.
Probability quantifies uncertainty by assigning numbers between 0 and 1 to events in a sample space.
Burnside's Lemma says the number of distinct objects up to a symmetry group equals the average number of objects fixed by each symmetry.
The partition function p(n) counts the number of ways to write n as a sum of positive integers where order does not matter.
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 the first kind count permutations by their number of cycles and connect power polynomials to rising/falling factorials.
Catalan numbers count many 'non-crossing' and 'well-formed' structures like balanced parentheses, binary trees, Dyck paths, and triangulations of a convex polygon.
Stirling numbers of the second kind S(n,k) count how many ways to split n labeled items into k non-empty, unlabeled groups.
A derangement is a permutation with no element left in its original position, often written as !n or D(n).
Lucas' Theorem lets you compute C(n, k) modulo a prime p by working digit-by-digit in base p.
The Inclusion-Exclusion Principle (IEP) corrects overcounting by alternately adding and subtracting sizes of intersections of sets.
A linear Diophantine equation ax + by = c has integer solutions if and only if gcd(a, b) divides c.