Schröder numbers: large S(n) counts lattice paths (0,0)→(n,n) using R=(1,0), U=(0,1), D=(1,1), never crossing y=x. S(n):1,2,6,22,90,394,1806... Small s(n)=S(n)/2 for n≥1: 1,1,3,11,45,197,903... Also called 'super-Catalan numbers'.
How to Use
Enter n:
- S(n): Large Schröder
- s(n): Small Schröder
- Sequence: First values
Catalan Relation
Catalan: paths with only R,U. Schröder: also allows D=(1,1). Delannoy: allows D but can cross diagonal. So C_n ≤ s(n) ≤ S(n) ≤ D(n,n). Each generalization adds more allowed paths.
Other Interpretations
S(n) also counts: (1) ways to parenthesize a sequence allowing empty groups, (2) dissections of a convex (n+2)-gon into polygons, (3) separable permutations of length n.
Step-by-Step Instructions
- 1Enter n.
- 2Compute S(n).
- 3Compute s(n).
- 4View sequence.
- 5Compare to Catalan.