Catalan numbers

The Catalan numbers (http://oeis.org/A000108) are probably the most ubiquitous sequence of integers in mathematics. They enumerate several important objects like binary trees, Dyck paths, and triangulations of convex polygons.

Main definitions

• catalan n: the nth Catalan number, defined recursively as catalan (n + 1) = ∑ i : Fin n.succ, catalan i * catalan (n - i).

Main results

• catalan_eq_centralBinom_div: The explicit formula for the Catalan number using the central binomial coefficient, catalan n = Nat.centralBinom n / (n + 1).

Implementation details

The proof of catalan_eq_centralBinom_div follows https://math.stackexchange.com/questions/3304415

TODO

• Prove that the Catalan numbers enumerate many interesting objects.
• Provide the many variants of Catalan numbers, e.g. associated to complex reflection groups, Fuss-Catalan, etc.

@[irreducible]

def catalan : ℕ → ℕ

The recursive definition of the sequence of Catalan numbers: catalan (n + 1) = ∑ i : Fin n.succ, catalan i * catalan (n - i)

Equations

• catalan 0 = 1
• catalan n.succ = ∑ i : Fin n.succ, catalan ↑i * catalan (n - ↑i)

@[simp]

theorem catalan_zero : catalan 0 = 1

theorem catalan_succ (n : ℕ) : catalan (n + 1) = ∑ i : Fin n.succ, catalan ↑i * catalan (n - ↑i)

theorem catalan_succ' (n : ℕ) : catalan (n + 1) = ∑ ij ∈ Finset.antidiagonal n, catalan ij.1 * catalan ij.2

@[simp]

theorem catalan_one : catalan 1 = 1

theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1)

theorem succ_mul_catalan_eq_centralBinom (n : ℕ) : (n + 1) * catalan n = n.centralBinom

theorem catalan_two : catalan 2 = 2

theorem catalan_three : catalan 3 = 5