Combinatorics Explorer

Counting principles, Pascal's patterns, and partition theory.

Perms & Combs

n (total items)

r (chosen items)

Pascal's Triangle

Rows (max 15)

Highlight Odd Numbers

Stirling & Bell

k (for Stirling)

Integer Partitions

n (max 40)

Catalan & Derangements

Generating Functions

Sequence

Show first N terms (≤20)

Ramsey Theory

No controls needed. View the known Ramsey numbers and results.

Pólya Enumeration

Beads n

Colors k

Binomial Theorem

Term a

Term b

Power n

Permutations P(n,r) and Combinations C(n,r)

P(n,r) ordered—

C(n,r) unordered—

Multiset C(n+r-1,r)—

n!—

r!—

(n-r)!—

Pascal's Triangle

Stirling & Bell Numbers

S(n,k) Stirling 2nd—ways to partition n into k non-empty subsets

Bell B(n)—total partitions of n-element set

[n,k] Stirling 1st—unsigned: perms of n with k cycles

Integer Partitions p(n)

p(n) = number of partitions —

Catalan Numbers & Derangements

Catalan Cₙ—C(2n,n)/(n+1)

Derangement D(n)—no fixed points permutations

Probability D/n!—→ 1/e ≈ 0.368

Generating Functions

A generating function encodes a sequence aₙ as coefficients of a power series: G(x) = Σ aₙxⁿ. Select a sequence to see its OGF.

Ramsey Theory

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.

Key Results

• R(3,3) = 6: Any 6-person party has 3 mutual friends or 3 mutual strangers
• R(4,4) = 18 (Exact, proven 1979)
• R(5,5): Known to be between 43 and 48 — open problem!
• Upper bound: R(s,t) ≤ R(s-1,t) + R(s,t-1)
• Probabilistic lower bound: R(k,k) ≥ √2 · 2^(k/2)

Pólya Enumeration Theorem

Counts distinct colorings of a set of objects up to symmetry using the cycle index of a group.

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Distinct Necklaces

Burnside's Lemma

|X/G| = (1/|G|) Σ|Fix(g)|. Number of distinct objects = average number of fixed points.

For n-bead necklaces with k colors under Cₙ rotations:

(1/n) Σ_{d|n} φ(d)·k^(n/d)

where φ is Euler's totient function.

Binomial Theorem

(a+b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ.

Combinatorics & Enumeration Handbook

Mathematical Principles & Theorems

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 \(!n = n!\sum_{i=0}^n \frac{(-1)^i}{i!} \approx \lfloor \frac{n!}{e} + \frac{1}{2} \rfloor\).

Operating Instructions

Select an enumeration formula (Permutations, Combinations, Catalan, Stirling, Derangements).
Enter parameters \(n\) and \(k\) into the numeric value boxes.
Click Calculate to view exact arbitrary-precision integer results.
Inspect the breakdown cards for combinatorial interpretations and recurrence equations.