Departure question
Find order without counting everythingPort 2 of 16Trace recurring structure through exploding possibilities, divisibility, and the apparent irregularity of primes.
This voyage is an editorial path for understanding, not a claim of direct historical influence or sole invention.
The same sequence appears in many counting problems that initially look unrelated. The number of balanced strings with n pairs of parentheses, full binary trees with n internal nodes, and triangulations of an (n+2)-gon is Cₙ. Bijections between those objects reveal much of the fun of combinatorics.
n | Cₙ | Meaning (example) |
|---|---|---|
| 0 | 1 | Empty expression |
| 1 | 1 | () |
| 2 | 2 | ()(), (()) |
| 3 | 5 | 5 balanced-parenthesis strings |
| 4 | 14 | 14 binary-tree shapes |
| 5 | 42 | 42 triangulations of a heptagon |
| 6 | 132 | |
| 7 | 429 | |
| 8 | 1430 |
One sequence answers dozens of seemingly different counting problems.
The same numbers appear in counting parentheses, trees, and lattice paths.
Cₙ = 1, 1, 2, 5, 14, 42, 132, ...
Ports in time
Follow the scenes to see problems, notation, standards of proof, and applications changing across different times and places.
Euler counted ways to triangulate a polygon, bringing the sequence 1, 2, 5, 14, 42, … into view.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearBelgian mathematician Eugène Catalan studied the same counting pattern through parenthesized products. The sequence later took his name.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearRichard Stanley assembled 214 distinct combinatorial interpretations of the Catalan numbers — remarkably different problems with the same answer sequence.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearCompiler parse trees, counting binary search trees, algorithm analysis, RNA secondary-structure prediction, and game trees.
Curated sources and problems. Bring one discovery back from OEIS, Project Euler, MathOverflow, or arXiv.
No concept belongs to one person
These are not inventor credits. They are different ports: opening a problem, sharpening a language, or carrying it into another world.
Number lenses
These numbers are editorial lenses for the voyage, not required prerequisites.
Trace recurring structure through exploding possibilities, divisibility, and the apparent irregularity of primes.
Open the number voyage
Trace recurring structure through exploding possibilities, divisibility, and the apparent irregularity of primes.
Open the number voyage
Trace recurring structure through exploding possibilities, divisibility, and the apparent irregularity of primes.
Open the number voyage
Concept genealogy
Concepts arriving from before
Current port
Catalan Numbers
Concepts opened from here
No direct successor port is curated yet.
Only direct editorial links are shown; this is not a complete learning order or historical influence line.
