# mathlibdocumentation

combinatorics.catalan

# 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_central_binom_div: The explicit formula for the Catalan number using the central binomial coefficient, catalan n = nat.central_binom n / (n + 1).

## Implementation details #

The proof of catalan_eq_central_binom_div follows https://math.stackexchange.com/questions/3304415/catalan-numbers-algebraic-proof-of-the-recurrence-relation

## 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.
def catalan  :

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

Equations
@[simp]
theorem catalan_zero  :
= 1
theorem catalan_succ (n : ) :
catalan (n + 1) = finset.univ.sum (λ (i : fin n.succ), * catalan (n - i))
@[simp]
theorem catalan_one  :
= 1
theorem catalan_two  :
= 2
theorem catalan_three  :
= 5