Narayana numbers: N(n,k) = (1/n)·C(n,k)·C(n,k-1). They refine Catalan: Σ_{k=1}^n N(n,k) = C_n. N(n,k) counts Dyck paths of semilength n with exactly k peaks (local maxima). The Narayana triangle is a beautiful combinatorial object.
How to Use
Enter n and k:
- N(n,k): Narayana number
- Triangle: Full display
- Row sum: = C_n
Narayana Triangle
Row 1: 1. Row 2: 1 1. Row 3: 1 3 1. Row 4: 1 6 6 1. Row 5: 1 10 20 10 1. Compare to Pascal's triangle — similar symmetry but different values. Each row sums to the Catalan number.
Combinatorics
N(n,k) also counts: (1) non-crossing partitions of [n] with k blocks. (2) Planar trees with n edges and k leaves. (3) Ways to triangulate a convex (n+2)-gon with k interior vertices on one side.
Step-by-Step Instructions
- 1Enter n.
- 2Enter k.
- 3Compute N(n,k).
- 4View triangle.
- 5Check Catalan sum.