Has this notation for functions on finite sets been used before?

Question

In group theory, specifically in groups of permutations, there is a special notation used for functions. I am wondering if a generalization of this notation has been used before. Let our finite set be $\{1,...,n\}$. Consider a word of length $n$ from that set. Let me give a specific example. Consider the set $\{1,2,3\}$. The word $112$ represents the function that outputs $1$ to the input $1$, $1$ to the input $2$, and $2$ to the input $3$. The word $232$ represents the function that outputs $2$ to the input $1$, $3$ to the input $2$, and $2$ to the input $3$. I hope the examples I have given make it sufficiently clear what the notation means. My question is, is there any book or paper where this notation has been used? Or am I the first one to use it?

Answer 1

This notation becomes problematic for $n \ge 10$ without at least some spacing, but it reminds me a lot of a notion I saw used for permutation groups. We could define a permutation $\sigma \in S_n$ with the notation $$ \left( \begin{matrix} 1 & 2 & 3 & \cdots & n \\ \sigma(1) & \sigma(2) & \sigma(3) & \cdots & \sigma(n) \end{matrix} \right) $$ i.e. the first row denotes your inputs and the second the outputs. Of course, this comes with the constraint (in the original context) of $\sigma$ being a bijection on $\{1,2,\cdots,n\}$, but there's no reason the notation could not be used here too.

Amusingly, this sent me on a quick hunt to find the name of the notation (Wikipedia just calls it "two line notation") -- and there is, in turn, apparently a "one line notation", used often enough to warrant inclusion on Wikipedia (link). So we could define the same $\sigma$ notationally by $$ \left( \begin{matrix} \sigma(1) & \sigma(2) & \sigma(3) & \cdots & \sigma(n) \end{matrix} \right) $$ This immediately clashes a bit with the cycle notation, though, wherein one might have $$ \left( \begin{matrix} \sigma(1) &\sigma(\sigma(1)) & \sigma(\sigma(\sigma(1))) & \cdots \end{matrix} \right) $$ stopping at some point at which point one loops to the start. That is, in cycle notation,

$$ \left( \begin{matrix} 1 & 3 & 2\end{matrix} \right) $$ represents a bijection $\sigma : \{1,2,3\} \to \{1,2,3\}$ defined by $$ \sigma(1) = 3 \qquad \sigma(3) = 2 \qquad \sigma(2) = 1 $$

Hence given the conflicting notations, one should probably omit the parentheses or use other sorts of bracketing for the one-line expression.

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In summary, your notation is much like an existing one in the study of permutations/symmetric groups (and should generalize to non-bijective functions just fine). It has some conflicts with an existing notation, however, and can be cumbersome if not written well.