Work out catalan path counting instantly with clear inputs, formula shown and shareable results.
On an n × n grid there are C(2n, n) monotonic paths from corner to corner. Exactly Cₙ = C(2n, n)/(n + 1) of them stay weakly below the diagonal, which is the classic Catalan interpretation.
Catalan paths
Cₙ = C(2n, n) / (n + 1)
All monotonic paths
C(2n, n)
924 monotonic paths in total, of which 132 stay below the diagonal.
Balanced bracket strings of length 2n, binary trees with n nodes and triangulations of an (n + 2)-gon — all the same number.