Counting principles, Pascal's patterns, and partition theory.
Enumerative combinatorics studies configurations and counting. Permutations without replacement \(P(n, k) = \frac{n!}{(n-k)!}\), Combinations \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), Catalan numbers \(C_n = \frac{1}{n+1}\binom{2n}{n}\) (counting Dyck paths and binary trees), Stirling numbers of the second kind \(S(n, k) = \frac{1}{k!}\sum_{j=0}^k (-1)^{k-j}\binom{k}{j}j^n\), and Derangements \(!n = n!\sum_{i=0}^n \frac{(-1)^i}{i!} \approx \lfloor \frac{n!}{e} + \frac{1}{2} \rfloor\).
No controls needed. View the known Ramsey numbers and results.
A generating function encodes a sequence aₙ as coefficients of a power series: G(x) = Σ aₙxⁿ. Select a sequence to see its OGF.
R(s,t): minimum n such that any 2-coloring of edges of Kₙ contains a red Kₛ or blue Kₜ. Known exact values are rare.
Counts distinct colorings of a set of objects up to symmetry using the cycle index of a group.
|X/G| = (1/|G|) Σ|Fix(g)|. Number of distinct objects = average number of fixed points.
(a+b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ.