How to Use
Enter n:
- Triangle: Full display
- Diagonal: Catalan numbers
- Row sums: Central binomials
Ballot Connection
C(n,k) counts the number of ways candidate A can maintain a lead throughout counting, with A getting n votes and B getting k votes (0≤k≤n). The Bertrand ballot problem pioneered this counting.
vs Pascal Triangle
Pascal: C(n,k) = C(n-1,k-1)+C(n-1,k). Catalan: different recurrence. But related: Catalan triangle entries = C(n+k,k)−C(n+k,k-1) (ballot numbers). Both are fundamental combinatorial triangles.
Step-by-Step Instructions
- 1Enter n.
- 2View triangle.
- 3Check diagonal.
- 4Verify Catalan.
- 5Compare to Pascal.
Catalan Triangle Calculator — Frequently Asked Questions
How does the diagonal give Catalan numbers?+
C(n,n) in the Catalan triangle equals the nth Catalan number. This is because C(n,n) counts complete ballot sequences where A is never behind — exactly the definition of Catalan via Dyck paths.
What are the row sums?+
Row sums Σ_k C(n,k) = C(2n,n) = central binomial coefficients. This is because every lattice path (0,0)→(n,n) can be classified by the first time it touches the diagonal, giving the Catalan triangle decomposition.
Are there other 'Catalan triangles'?+
Yes, several! Different recurrences produce different triangles all connected to Catalan numbers. The most common: ballot numbers, Shapiro's Catalan triangle, and Aigner's Catalan triangle. All have Catalan numbers on a diagonal or edge.
More Calculators Tools
Explore all 639 other calculators tools available on Generatr