# Mathematical Notation of Sequence of Functions

Let's say I have a finite set of functions $F=\{f_1,f_2,f_3,...,f_n\}$ and I want to show a recursive function that is constructed by an arbitrary sequence of applications of functions in $F$ to input $x$. For example, one such sequence could be $f_4(f_9(f_1(x)))$. Is there a nice mathematical way to show the set of all possible sequences of applying elements of $F$ to input $x$?

• You need to clarify: Do you want only permutations of the set applied to x or any sequence of the set applied to x? Your example is not does not feature a permutation of the set of functions. Commented Apr 30, 2016 at 23:01
• I think I should change that to sequence. Basically to mathematically show all the possible sequences of applications of functions in F to input X. (you could assume that the sequence is no longer than k so that it is a finite sequence) Commented Apr 30, 2016 at 23:07
• Then perhaps $\{(g_m \circ g_{m-1} \circ \ldots \circ g_1)(x) | m\leq k; g_i \in F \forall i=1,\ldots, m\}$ Commented Apr 30, 2016 at 23:11
• Nice! This also covers repetition right? for example in one sequence $g_1$ can be applied any arbitrary times? Commented Apr 30, 2016 at 23:12

What you could write is just

$$\{(g_m \circ g_{m-1} \circ \ldots \circ g_1)(x) | m\leq k; g_i \in F\, \forall i=1,\ldots,m\}$$

This does cover repetitions and allows any sequence of $f_i$'s up to length $k$.

A possibly related concept is that of an Iterated Function System, and the Hutchinson operator: $$H(S) = \bigcup_{i=1}^n f_i(S)$$.

In this setting, the collection of all iterates of your collection of functions $\{f_i\}$ applied to $x$ could be described as $$\bigcup_{j=0}^\infty H^{j}(\{x\})$$ where the exponent represents repeated applications of the Hutchinson operator.

Just a suggestion..

Usually $\sigma(A)$ denotes the set of all possible permutations of elements of $A$

To simplify you could write $\sigma(n) = \sigma(\{1, \dots, n\})$ and write then $f_{\sigma(n)}$ the set of all functions you describe

Or if you mean also subsequence (i.e. Permutations shorter than $n$) you could define $\sigma(n)= \cup_{i = 1}^n \sigma(\{1,\dots,i\})$

• My first version of the question was wrong in that it didn't cover repetition. For example $f_1$ can come more than once in the sequence. This changes the definition a bit. Is there a variation of your answer that can be applied to it? Commented Apr 30, 2016 at 23:14

Here it is a recursive definition of $\Gamma(x)$, the orbit of $x$:

Let $\Gamma_1(x)=\{f_1(x),\ldots,f_n(x)\}$.

Let $\Gamma_{m+1}(x)=\{ f_1(y),\ldots,f_n(y) / y\in \Gamma_{m}(x)\}$.

Finally, $$\Gamma(x)=\bigcup_{m\geq 1}\Gamma_m(x).$$