How to denote an ordered list, indicating that the order is not known beforehand

Question

I have a set of states $A = ${$ a_0, a_1, ..., a_n $}. A model predicts $y$, an ordered list in which these states were observed over time, e.g. $y = (a_3, a_1, a_4)$. Not every $a_i$ in $A$ needs to be observed, so $|y|$ is not necessarily $|A|$.

Now I am getting stuck on how to declare $y$ clearly.

I thought about $y = (a_i, a_{i+1}, ..., a_m), a\in A$, but I feel like this notation forces $a_3$ to be after $a_2$, which should really not be the case. But the notation should show that $y$ is ordered based on the times at which the states were observed, in this case $a_i$ being the first and $a_m$ being the last.

Any help on this would greatly be appreciated. I hope the question is clear, otherwise I am happy to clarify.

Answer 1

One way is to introduce a function $w$ that iterates over the desired / observed indices:
$$w:\{1,2,...,m\}\to \{1,2,...,n\}$$

so that the $m$-tuple of observed states is

$$y = (a_{w(1)}, a_{w(2)}, ..., a_{w(m)})$$

$w$ need not be surjective because not every state will occur in $y$, but it must be injective because no state will occur more than once: $$i\neq j\implies w(i)\neq w(j)$$

On notation: For functions on discrete domain, indexed notation like $w_1$ etc. is not uncommon. But as $w$ itself appears as indices, this might have negative impact on legibility: $y = (a_{w_1}, a_{w_2}, ..., a_{w_m})$. Therefore I used notation $w(i)$.

Answer 2

You could go with $$y=(a_{\sigma(1)},a_{\sigma(2)},\dots,a_{\sigma(m)})$$ for some function $\sigma : \{1,2,\dots m\}\to \{1,2,\dots, n\}$ where $m$ is the number of observed states. If a state cannot be observed twice then you need to force $\sigma$ to be injective.

Answer 3

You could also write "$y = (y_1, y_2, \ldots, y_m)$ where $y_i \in A.$" The idea here is that since the members of $A$ were originally enumerated by $a_1,$ $a_2,$ and so forth, we use a different letter to name the elements of the sequence. This notation gives no particular indication of any relationship between the sequence of elements $y_i$ and the sequence of the original enumeration $a_j.$

If your intention is that no state should ever be observed twice within the sequence, you would also have to specify something like

$$ i \neq j \implies y_i \neq y_j, $$

similar to the way in which another answer specifies that its index function $w(i)$ is injective. But in that case an advantage of defining an index function is that you can simply say "the injective function $w: \{1,2,\ldots,m\}\to \{1,2,\ldots,n\}$," where the single word "injective" saves you the trouble of writing out "$ i \neq j \implies w(i) \neq w(j) $."