Public index · Mathematics
Fifth Catalan number
42
C₅ in the Catalan sequence
Picture pairing parentheses so they always close. Or drawing paths that never go below the ground. The number of those ways, for a given size, is a Catalan number. The fifth one is 42. The one before it is 14. That pairing was noticed afterward.
Combinatorics · 1838
C₅=42
Picture pairing parentheses so they always close. Or drawing paths that never go below the ground. Or cutting a polygon into triangles without crossing. The number of ways to do those things, for a given size, is a Catalan number. Eugène Catalan wrote the sequence in 1838. The fifth one is 42. The one before it is 14. Fourteen and forty-two sit next to each other. Nobody named them for this Index. Lists reach forty-two on their own.
What it supports
A named combinatorial sequence. Eugène Catalan, 1838, though Euler had counted polygon triangulations earlier. Cₙ = (1/(n+1)) · (2n choose n). The sequence goes 1, 1, 2, 5, 14, 42, 132. C₄ is 14 noncrossing partitions of four points. C₅ is 42 of five. Exact inside one named sequence. Filed as calculated, post-hoc. Never as architecture rationale, and never as proof.
What it does not
Not a physical measurement. Not a rationale for fourteen movements or forty-two houses. Not the cake number, which counts plane cuts of a cube. Not the primary-pseudoperfect identity. Exact inside one named sequence among many sequences.
Calculation
C₅ = 1/6 · (10 choose 5) = 1/6 · 252 = 42
Catalan number formula
ExactnessExact
Form42
Unitinteger
Strengthhard
Source
OEIS A000108 · Catalan numbers- Stanley, Enumerative Combinatorics
- The Book of 42 · mathematics