MathDB

A polynomial

P

n

variables and real coefficients is called magical if

P(\mathbb{N}^n)\subset \mathbb{N}

, and moreover the map

P: \mathbb{N}^n \to \mathbb{N}

is a bijection. Prove that for all positive integers

n

, there are at least

n!\cdot (C(n)-C(n-1))

magical polynomials, where

C(n)

is the

n

-th Catalan number.
Here

\mathbb{N}=\{0,1,2,\dots\}

P1