Notation for the set of all injections from $A$ into $B$

The notation suggested by cardinal equalities such as

\begin{array}{l|l|l} \text{concept} & \text{notation} & \text{cardinality} \\ \hline \text{disjoint union of $A$ and $B$} & A + B & |A + B| = |A| + |B| \\ \text{Cartesian product of $A$ and $B$} & A \times B & |A \times B| = |A| \times |B| \\ \text{set of functions from $A$ to $B$, also $A \rightarrow B$} & B^A & |B^A| = |B|^{|A|} \\ \text{set of permutations of $A$, also $\text{Sym}(A)$} & A! & |A!| = |A|! \\ \text{set of $k$-element subsets of $A$} & \binom{A}{k} & \left|\binom{A}{k}\right| = \binom{|A|}{k} \\ \text{set of $k$-element partitions of $A$} & \left\{{A \atop k}\right\} & \left| \left\{{A \atop k}\right\} \right| = \left\{{|A| \atop k}\right\} \end{array}

is

\begin{array}{l|l|l} \text{concept} & \text{notation} & \text{cardinality} \\ \hline \text{set of injections from $A$ to $B$} & B^{\underline{A}} & |B^{\underline{A}}| = |B|^{\underline{|A|}} \end{array}

because the falling factorial

\begin{align*} |B|^\underline{|A|} = \frac{|B|!}{(|B| - |A|)!} \end{align*}

is precisely the number of injections from $A$ to $B$.

If I take your question literally, it seems to me that the correct answer is simply "No". But I quite happily use $B^A_{\neq}$. Analogously, if $A$ and $B$ happen to be ordered, I write $B^A_{<}$ for the set of all strictly increasing functions from $A$ to $B$. For me, this works well.