Cardinality of Set of Strictly Increasing Mappings

Theorem
Let $$\left({S, \preceq}\right)$$ and $$\left({T, \preccurlyeq}\right)$$ be tosets.

Let the cardinality of $$S$$ and $$T$$ be:
 * $$\left|{S}\right| = m, \left|{T}\right| = n$$

Then the number of strictly increasing mappings from $$S$$ to $$T$$ is $$\binom n m = \frac {n!} {m! \left({n - m}\right)!}$$.


 * $$\binom n m$$ is of course a binomial coefficient.

Proof
From Order Monomorphism iff Strictly Increasing and Strictly Monotone Mapping is Injective, a strictly increasing mapping $$\phi$$ from $$S$$ to $$T$$ is an order isomorphism from $$S$$ to $$\phi \left({S}\right)$$.

Let $$\mathbb F$$ be the set of all strictly increasing mappings from $$S$$ to $$T$$.

Let $$\mathbb G$$ be the set of all subsets of $$T$$ with $$m$$ elements.

By Unique Isomorphism between Finite Totally Ordered Sets, the mapping $$\Phi: \mathbb F \to \mathbb G$$ defined as:
 * $$\forall \phi \in \mathbb F: \Phi: \phi \to \phi \left({S}\right)$$

is a bijection.

The result follows from Cardinality of Set of Subsets.