# Invariant functions on the space of finite sequences of reals

Let $S$ be a space of all finite sequences of real numbers (we don't endow it with metric or topology in general). Before asking the main question, some notation.

1. For each $\mathbf s\in S$ we define $|\mathbf s|$ to be the length of $\mathbf s$.

2. Let $\mathbf s',\mathbf s''\in S$ be equivalent ($\mathbf s'\sim \mathbf s''$) if $|\mathbf s'| = |\mathbf s''|$ and $\mathbf s''$ can be obtained by $\mathbf s'$ by only permuting elements of $\mathbf s'$. Say, $\{1,2,3\}\sim\{2,3,1\}$.

3. Define an addition of sequences in the following way: $$\mathbf s = \mathbf s'+\mathbf s'' = \{s'_1,...,s'_{k},s''_1,...,s''_{l}\}.$$ where $k=|\mathbf s'|$ and $l = |\mathbf s''|$ and multiplication by scalar: $$\alpha\mathbf s = \{\alpha s_1,...,\alpha s_{|s|}\}.$$

4. For $n\in\mathbb N$ put $\mathbf 1_n = \{1,1,...,1\}$ such that $|\mathbf 1_n|=n$.

In probability theory often the following function $L\to \mathbb R$ are used: $$\min \mathbf s = \min_{1\leq i\leq |s|}s_i,$$ $$\max \mathbf s = \max_{1\leq i\leq |s|}s_i,$$ $$\overline{\mathbf s} = \frac{1}{|\mathbf s|}\sum s_i.$$

All these functions have a nice invariance property: $f(\mathbf s) = f\left(\mathbf s+f(\mathbf s)\mathbf 1_n\right)$ for all $n\in\mathbb N$. I am interested if this class of functions was already discussed in details?

Some thoughts:

a) if we put $S$ to be a class of real-valued continuous maps with compact domains from $\mathbb R$ then we can extend these three functions to admit the same property. Moreover, the outcome for each of these functions will be in an image of $\mathbf s$.

b) all these three functions are constant on class of equivalence $[\mathbf s]$ for any $\mathbf s\in S$.

c) For sure, $\overline{\mathbf s}$ is an expectation with respect to class of uniform probability measures each of them defined over a finite set. So, we can extend it to a wider class of measures. On the other hand, then we will lose the property b).

d) Constant function of course also admit these property, however is not of too much interest.

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@joriki: thanks for boldface. Also added d) and continuity in a). –  Ilya Jul 22 '11 at 11:39
You're welcome -- I added some more :-) –  joriki Jul 22 '11 at 11:45
A few more examples: Assuming that $\mathbf{s}$ is listed in non-decreasing order and that $n=|\mathbf{s}|$, define $r_k(\mathbf{s})=s_k$ if $k\le|\mathbf{s}|$, $r_k(\mathbf{s})=s_n$ otherwise; then $r_k$ has the property, as do the corresponding ‘rank from the top’ functions and the median. If $f:\mathbb{R}^2 \to \mathbb{R}$ satisfies $\min\{x,y\}\le f(\{x,y\})\le$ $\max \{x,y \}$, then $\mathbf{s}\mapsto f(\min\mathbf{s},\max\mathbf{s})$ has the property. All of these examples satisfy $f(\mathbf{s'})=\mathbf{s}$ whenever $\mathbf{s'}$ is the sum of finitely many copies of $\mathbf{s}$. –  Brian M. Scott Jul 23 '11 at 9:09
I do believe that the any of the power means (en.wikipedia.org/wiki/Power_mean) would be invariant (special cases for $-\infty, \infty$ and $1$ already mentioned). I believe you can find many more invariant functions as other means (median has been mentioned, the most common number with some way of handling a tie, and so on) –  Arthur Oct 1 '11 at 3:18
Note also that if this holds for n=1 it holds for all n. –  Harry Altman Oct 3 '11 at 2:53