The Catalan Numbers

• what are they - dyck paths

• simple examples

• recurrence and proof

  • C n+1 ←→ +/(C×(C n-⊢))¨0,⍳n

  • C←{⍵=0: 1 ⋄ +/(∇×(∇⍵-1+⊢))¨¯1+⍳⍵}

  • Cs←{(⊢,⊢+.×⌽)⍣(⍵-1),1}

• closed form and proof

  • C n ←→ (n!2×n)-(n-1)!2×n ←→ (n+1)÷⍨n!2×n

• more catalan number examples

• mention stanley book