Cardinality of Set of Strictly Increasing Mappings

Theorem
Let $\struct {S, \preceq}$ and $\struct {T, \preccurlyeq}$ be tosets.

Let the cardinality of $S$ and $T$ be:
 * $\card S = m, \card T = n$

Then the number of strictly increasing mappings from $S$ to $T$ is:
 * $\dbinom n m = \dfrac {n!} {m! \ \paren {n - m}!}$.

where $\dbinom n m$ is a binomial coefficient.

Proof
From:
 * Mapping from Totally Ordered Set is Order Embedding iff Strictly Increasing

and:
 * Strictly Monotone Mapping with Totally Ordered Domain is Injective

a strictly increasing mapping $\phi$ from $S$ to $T$ is an order isomorphism from $S$ to $\map \phi S$.

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 Equivalent Finite Totally Ordered Sets, the mapping $\Phi: \mathbb F \to \mathbb G$ defined as:
 * $\forall \phi \in \mathbb F: \Phi: \phi \to \map \phi S$

is a bijection.

The result follows from Cardinality of Set of Subsets.